<!DOCTYPE html><html><head>
      <title>03</title>
      <meta charset="utf-8">
      <meta name="viewport" content="width=device-width, initial-scale=1.0">
      
      <style>
      /*!
 * reveal.js
 * http://revealjs.com
 * MIT licensed
 *
 * Copyright (C) 2018 Hakim El Hattab, http://hakim.se
 */
/*********************************************
 * RESET STYLES
 *********************************************/
html, body, .reveal div, .reveal span, .reveal applet, .reveal object, .reveal iframe,
.reveal h1, .reveal h2, .reveal h3, .reveal h4, .reveal h5, .reveal h6, .reveal p, .reveal blockquote, .reveal pre,
.reveal a, .reveal abbr, .reveal acronym, .reveal address, .reveal big, .reveal cite, .reveal code,
.reveal del, .reveal dfn, .reveal em, .reveal img, .reveal ins, .reveal kbd, .reveal q, .reveal s, .reveal samp,
.reveal small, .reveal strike, .reveal strong, .reveal sub, .reveal sup, .reveal tt, .reveal var,
.reveal b, .reveal u, .reveal center,
.reveal dl, .reveal dt, .reveal dd, .reveal ol, .reveal ul, .reveal li,
.reveal fieldset, .reveal form, .reveal label, .reveal legend,
.reveal table, .reveal caption, .reveal tbody, .reveal tfoot, .reveal thead, .reveal tr, .reveal th, .reveal td,
.reveal article, .reveal aside, .reveal canvas, .reveal details, .reveal embed,
.reveal figure, .reveal figcaption, .reveal footer, .reveal header, .reveal hgroup,
.reveal menu, .reveal nav, .reveal output, .reveal ruby, .reveal section, .reveal summary,
.reveal time, .reveal mark, .reveal audio, .reveal video {
  margin: 0;
  padding: 0;
  border: 0;
  font-size: 100%;
  /* font: inherit; */
  vertical-align: baseline; }

.reveal article, .reveal aside, .reveal details, .reveal figcaption, .reveal figure,
.reveal footer, .reveal header, .reveal hgroup, .reveal menu, .reveal nav, .reveal section {
  display: block; }

/*********************************************
 * GLOBAL STYLES
 *********************************************/
html,
body {
  width: 100%;
  height: 100%;
  overflow: hidden; }

body {
  position: relative;
  line-height: 1;
  background-color: #fff;
  color: #000; }

/*********************************************
 * VIEW FRAGMENTS
 *********************************************/
.reveal .slides section .fragment {
  opacity: 0;
  visibility: hidden;
  transition: all .2s ease; }
  .reveal .slides section .fragment.visible {
    opacity: 1;
    visibility: inherit; }

.reveal .slides section .fragment.grow {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.grow.visible {
    -webkit-transform: scale(1.3);
            transform: scale(1.3); }

.reveal .slides section .fragment.shrink {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.shrink.visible {
    -webkit-transform: scale(0.7);
            transform: scale(0.7); }

.reveal .slides section .fragment.zoom-in {
  -webkit-transform: scale(0.1);
          transform: scale(0.1); }
  .reveal .slides section .fragment.zoom-in.visible {
    -webkit-transform: none;
            transform: none; }

.reveal .slides section .fragment.fade-out {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.fade-out.visible {
    opacity: 0;
    visibility: hidden; }

.reveal .slides section .fragment.semi-fade-out {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.semi-fade-out.visible {
    opacity: 0.5;
    visibility: inherit; }

.reveal .slides section .fragment.strike {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.strike.visible {
    text-decoration: line-through; }

.reveal .slides section .fragment.fade-up {
  -webkit-transform: translate(0, 20%);
          transform: translate(0, 20%); }
  .reveal .slides section .fragment.fade-up.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-down {
  -webkit-transform: translate(0, -20%);
          transform: translate(0, -20%); }
  .reveal .slides section .fragment.fade-down.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-right {
  -webkit-transform: translate(-20%, 0);
          transform: translate(-20%, 0); }
  .reveal .slides section .fragment.fade-right.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-left {
  -webkit-transform: translate(20%, 0);
          transform: translate(20%, 0); }
  .reveal .slides section .fragment.fade-left.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-in-then-out,
.reveal .slides section .fragment.current-visible {
  opacity: 0;
  visibility: hidden; }
  .reveal .slides section .fragment.fade-in-then-out.current-fragment,
  .reveal .slides section .fragment.current-visible.current-fragment {
    opacity: 1;
    visibility: inherit; }

.reveal .slides section .fragment.fade-in-then-semi-out {
  opacity: 0;
  visibility: hidden; }
  .reveal .slides section .fragment.fade-in-then-semi-out.visible {
    opacity: 0.5;
    visibility: inherit; }
  .reveal .slides section .fragment.fade-in-then-semi-out.current-fragment {
    opacity: 1;
    visibility: inherit; }

.reveal .slides section .fragment.highlight-red,
.reveal .slides section .fragment.highlight-current-red,
.reveal .slides section .fragment.highlight-green,
.reveal .slides section .fragment.highlight-current-green,
.reveal .slides section .fragment.highlight-blue,
.reveal .slides section .fragment.highlight-current-blue {
  opacity: 1;
  visibility: inherit; }

.reveal .slides section .fragment.highlight-red.visible {
  color: #ff2c2d; }

.reveal .slides section .fragment.highlight-green.visible {
  color: #17ff2e; }

.reveal .slides section .fragment.highlight-blue.visible {
  color: #1b91ff; }

.reveal .slides section .fragment.highlight-current-red.current-fragment {
  color: #ff2c2d; }

.reveal .slides section .fragment.highlight-current-green.current-fragment {
  color: #17ff2e; }

.reveal .slides section .fragment.highlight-current-blue.current-fragment {
  color: #1b91ff; }

/*********************************************
 * DEFAULT ELEMENT STYLES
 *********************************************/
/* Fixes issue in Chrome where italic fonts did not appear when printing to PDF */
.reveal:after {
  content: '';
  font-style: italic; }

.reveal iframe {
  z-index: 1; }

/** Prevents layering issues in certain browser/transition combinations */
.reveal a {
  position: relative; }

.reveal .stretch {
  max-width: none;
  max-height: none; }

.reveal pre.stretch code {
  height: 100%;
  max-height: 100%;
  box-sizing: border-box; }

/*********************************************
 * CONTROLS
 *********************************************/
@-webkit-keyframes bounce-right {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateX(0);
            transform: translateX(0); }
  20% {
    -webkit-transform: translateX(10px);
            transform: translateX(10px); }
  30% {
    -webkit-transform: translateX(-5px);
            transform: translateX(-5px); } }
@keyframes bounce-right {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateX(0);
            transform: translateX(0); }
  20% {
    -webkit-transform: translateX(10px);
            transform: translateX(10px); }
  30% {
    -webkit-transform: translateX(-5px);
            transform: translateX(-5px); } }

@-webkit-keyframes bounce-down {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateY(0);
            transform: translateY(0); }
  20% {
    -webkit-transform: translateY(10px);
            transform: translateY(10px); }
  30% {
    -webkit-transform: translateY(-5px);
            transform: translateY(-5px); } }

@keyframes bounce-down {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateY(0);
            transform: translateY(0); }
  20% {
    -webkit-transform: translateY(10px);
            transform: translateY(10px); }
  30% {
    -webkit-transform: translateY(-5px);
            transform: translateY(-5px); } }

.reveal .controls {
  display: none;
  position: absolute;
  top: auto;
  bottom: 12px;
  right: 12px;
  left: auto;
  z-index: 1;
  color: #000;
  pointer-events: none;
  font-size: 10px; }
  .reveal .controls button {
    position: absolute;
    padding: 0;
    background-color: transparent;
    border: 0;
    outline: 0;
    cursor: pointer;
    color: currentColor;
    -webkit-transform: scale(0.9999);
            transform: scale(0.9999);
    transition: color 0.2s ease, opacity 0.2s ease, -webkit-transform 0.2s ease;
    transition: color 0.2s ease, opacity 0.2s ease, transform 0.2s ease;
    z-index: 2;
    pointer-events: auto;
    font-size: inherit;
    visibility: hidden;
    opacity: 0;
    -webkit-appearance: none;
    -webkit-tap-highlight-color: transparent; }
  .reveal .controls .controls-arrow:before,
  .reveal .controls .controls-arrow:after {
    content: '';
    position: absolute;
    top: 0;
    left: 0;
    width: 2.6em;
    height: 0.5em;
    border-radius: 0.25em;
    background-color: currentColor;
    transition: all 0.15s ease, background-color 0.8s ease;
    -webkit-transform-origin: 0.2em 50%;
            transform-origin: 0.2em 50%;
    will-change: transform; }
  .reveal .controls .controls-arrow {
    position: relative;
    width: 3.6em;
    height: 3.6em; }
    .reveal .controls .controls-arrow:before {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(45deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(45deg); }
    .reveal .controls .controls-arrow:after {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-45deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(-45deg); }
    .reveal .controls .controls-arrow:hover:before {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(40deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(40deg); }
    .reveal .controls .controls-arrow:hover:after {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-40deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(-40deg); }
    .reveal .controls .controls-arrow:active:before {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(36deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(36deg); }
    .reveal .controls .controls-arrow:active:after {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-36deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(-36deg); }
  .reveal .controls .navigate-left {
    right: 6.4em;
    bottom: 3.2em;
    -webkit-transform: translateX(-10px);
            transform: translateX(-10px); }
  .reveal .controls .navigate-right {
    right: 0;
    bottom: 3.2em;
    -webkit-transform: translateX(10px);
            transform: translateX(10px); }
    .reveal .controls .navigate-right .controls-arrow {
      -webkit-transform: rotate(180deg);
              transform: rotate(180deg); }
    .reveal .controls .navigate-right.highlight {
      -webkit-animation: bounce-right 2s 50 both ease-out;
              animation: bounce-right 2s 50 both ease-out; }
  .reveal .controls .navigate-up {
    right: 3.2em;
    bottom: 6.4em;
    -webkit-transform: translateY(-10px);
            transform: translateY(-10px); }
    .reveal .controls .navigate-up .controls-arrow {
      -webkit-transform: rotate(90deg);
              transform: rotate(90deg); }
  .reveal .controls .navigate-down {
    right: 3.2em;
    bottom: 0;
    -webkit-transform: translateY(10px);
            transform: translateY(10px); }
    .reveal .controls .navigate-down .controls-arrow {
      -webkit-transform: rotate(-90deg);
              transform: rotate(-90deg); }
    .reveal .controls .navigate-down.highlight {
      -webkit-animation: bounce-down 2s 50 both ease-out;
              animation: bounce-down 2s 50 both ease-out; }
  .reveal .controls[data-controls-back-arrows="faded"] .navigate-left.enabled,
  .reveal .controls[data-controls-back-arrows="faded"] .navigate-up.enabled {
    opacity: 0.3; }
    .reveal .controls[data-controls-back-arrows="faded"] .navigate-left.enabled:hover,
    .reveal .controls[data-controls-back-arrows="faded"] .navigate-up.enabled:hover {
      opacity: 1; }
  .reveal .controls[data-controls-back-arrows="hidden"] .navigate-left.enabled,
  .reveal .controls[data-controls-back-arrows="hidden"] .navigate-up.enabled {
    opacity: 0;
    visibility: hidden; }
  .reveal .controls .enabled {
    visibility: visible;
    opacity: 0.9;
    cursor: pointer;
    -webkit-transform: none;
            transform: none; }
  .reveal .controls .enabled.fragmented {
    opacity: 0.5; }
  .reveal .controls .enabled:hover,
  .reveal .controls .enabled.fragmented:hover {
    opacity: 1; }

.reveal:not(.has-vertical-slides) .controls .navigate-left {
  bottom: 1.4em;
  right: 5.5em; }

.reveal:not(.has-vertical-slides) .controls .navigate-right {
  bottom: 1.4em;
  right: 0.5em; }

.reveal:not(.has-horizontal-slides) .controls .navigate-up {
  right: 1.4em;
  bottom: 5em; }

.reveal:not(.has-horizontal-slides) .controls .navigate-down {
  right: 1.4em;
  bottom: 0.5em; }

.reveal.has-dark-background .controls {
  color: #fff; }

.reveal.has-light-background .controls {
  color: #000; }

.reveal.no-hover .controls .controls-arrow:hover:before,
.reveal.no-hover .controls .controls-arrow:active:before {
  -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(45deg);
          transform: translateX(0.5em) translateY(1.55em) rotate(45deg); }

.reveal.no-hover .controls .controls-arrow:hover:after,
.reveal.no-hover .controls .controls-arrow:active:after {
  -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-45deg);
          transform: translateX(0.5em) translateY(1.55em) rotate(-45deg); }

@media screen and (min-width: 500px) {
  .reveal .controls[data-controls-layout="edges"] {
    top: 0;
    right: 0;
    bottom: 0;
    left: 0; }
  .reveal .controls[data-controls-layout="edges"] .navigate-left,
  .reveal .controls[data-controls-layout="edges"] .navigate-right,
  .reveal .controls[data-controls-layout="edges"] .navigate-up,
  .reveal .controls[data-controls-layout="edges"] .navigate-down {
    bottom: auto;
    right: auto; }
  .reveal .controls[data-controls-layout="edges"] .navigate-left {
    top: 50%;
    left: 8px;
    margin-top: -1.8em; }
  .reveal .controls[data-controls-layout="edges"] .navigate-right {
    top: 50%;
    right: 8px;
    margin-top: -1.8em; }
  .reveal .controls[data-controls-layout="edges"] .navigate-up {
    top: 8px;
    left: 50%;
    margin-left: -1.8em; }
  .reveal .controls[data-controls-layout="edges"] .navigate-down {
    bottom: 8px;
    left: 50%;
    margin-left: -1.8em; } }

/*********************************************
 * PROGRESS BAR
 *********************************************/
.reveal .progress {
  position: absolute;
  display: none;
  height: 3px;
  width: 100%;
  bottom: 0;
  left: 0;
  z-index: 10;
  background-color: rgba(0, 0, 0, 0.2);
  color: #fff; }

.reveal .progress:after {
  content: '';
  display: block;
  position: absolute;
  height: 10px;
  width: 100%;
  top: -10px; }

.reveal .progress span {
  display: block;
  height: 100%;
  width: 0px;
  background-color: currentColor;
  transition: width 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985); }

/*********************************************
 * SLIDE NUMBER
 *********************************************/
.reveal .slide-number {
  position: absolute;
  display: block;
  right: 8px;
  bottom: 8px;
  z-index: 31;
  font-family: Helvetica, sans-serif;
  font-size: 12px;
  line-height: 1;
  color: #fff;
  background-color: rgba(0, 0, 0, 0.4);
  padding: 5px; }

.reveal .slide-number a {
  color: currentColor; }

.reveal .slide-number-delimiter {
  margin: 0 3px; }

/*********************************************
 * SLIDES
 *********************************************/
.reveal {
  position: relative;
  width: 100%;
  height: 100%;
  overflow: hidden;
  -ms-touch-action: none;
      touch-action: none; }

@media only screen and (orientation: landscape) {
  .reveal.ua-iphone {
    position: fixed; } }

.reveal .slides {
  position: absolute;
  width: 100%;
  height: 100%;
  top: 0;
  right: 0;
  bottom: 0;
  left: 0;
  margin: auto;
  pointer-events: none;
  overflow: visible;
  z-index: 1;
  text-align: center;
  -webkit-perspective: 600px;
          perspective: 600px;
  -webkit-perspective-origin: 50% 40%;
          perspective-origin: 50% 40%; }

.reveal .slides > section {
  -ms-perspective: 600px; }

.reveal .slides > section,
.reveal .slides > section > section {
  display: none;
  position: absolute;
  width: 100%;
  padding: 20px 0px;
  pointer-events: auto;
  z-index: 10;
  -webkit-transform-style: flat;
          transform-style: flat;
  transition: -webkit-transform-origin 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), -webkit-transform 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), visibility 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), opacity 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985);
  transition: transform-origin 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), transform 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), visibility 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), opacity 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985); }

/* Global transition speed settings */
.reveal[data-transition-speed="fast"] .slides section {
  transition-duration: 400ms; }

.reveal[data-transition-speed="slow"] .slides section {
  transition-duration: 1200ms; }

/* Slide-specific transition speed overrides */
.reveal .slides section[data-transition-speed="fast"] {
  transition-duration: 400ms; }

.reveal .slides section[data-transition-speed="slow"] {
  transition-duration: 1200ms; }

.reveal .slides > section.stack {
  padding-top: 0;
  padding-bottom: 0;
  pointer-events: none; }

.reveal .slides > section.present,
.reveal .slides > section > section.present {
  display: block;
  z-index: 11;
  opacity: 1; }

.reveal .slides > section:empty,
.reveal .slides > section > section:empty,
.reveal .slides > section[data-background-interactive],
.reveal .slides > section > section[data-background-interactive] {
  pointer-events: none; }

.reveal.center,
.reveal.center .slides,
.reveal.center .slides section {
  min-height: 0 !important; }

/* Don't allow interaction with invisible slides */
.reveal .slides > section.future,
.reveal .slides > section > section.future,
.reveal .slides > section.past,
.reveal .slides > section > section.past {
  pointer-events: none; }

.reveal.overview .slides > section,
.reveal.overview .slides > section > section {
  pointer-events: auto; }

.reveal .slides > section.past,
.reveal .slides > section.future,
.reveal .slides > section > section.past,
.reveal .slides > section > section.future {
  opacity: 0; }

/*********************************************
 * Mixins for readability of transitions
 *********************************************/
/*********************************************
 * SLIDE TRANSITION
 * Aliased 'linear' for backwards compatibility
 *********************************************/
.reveal.slide section {
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden; }

.reveal .slides > section[data-transition=slide].past,
.reveal .slides > section[data-transition~=slide-out].past,
.reveal.slide .slides > section:not([data-transition]).past {
  -webkit-transform: translate(-150%, 0);
          transform: translate(-150%, 0); }

.reveal .slides > section[data-transition=slide].future,
.reveal .slides > section[data-transition~=slide-in].future,
.reveal.slide .slides > section:not([data-transition]).future {
  -webkit-transform: translate(150%, 0);
          transform: translate(150%, 0); }

.reveal .slides > section > section[data-transition=slide].past,
.reveal .slides > section > section[data-transition~=slide-out].past,
.reveal.slide .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate(0, -150%);
          transform: translate(0, -150%); }

.reveal .slides > section > section[data-transition=slide].future,
.reveal .slides > section > section[data-transition~=slide-in].future,
.reveal.slide .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate(0, 150%);
          transform: translate(0, 150%); }

.reveal.linear section {
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden; }

.reveal .slides > section[data-transition=linear].past,
.reveal .slides > section[data-transition~=linear-out].past,
.reveal.linear .slides > section:not([data-transition]).past {
  -webkit-transform: translate(-150%, 0);
          transform: translate(-150%, 0); }

.reveal .slides > section[data-transition=linear].future,
.reveal .slides > section[data-transition~=linear-in].future,
.reveal.linear .slides > section:not([data-transition]).future {
  -webkit-transform: translate(150%, 0);
          transform: translate(150%, 0); }

.reveal .slides > section > section[data-transition=linear].past,
.reveal .slides > section > section[data-transition~=linear-out].past,
.reveal.linear .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate(0, -150%);
          transform: translate(0, -150%); }

.reveal .slides > section > section[data-transition=linear].future,
.reveal .slides > section > section[data-transition~=linear-in].future,
.reveal.linear .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate(0, 150%);
          transform: translate(0, 150%); }

/*********************************************
 * CONVEX TRANSITION
 * Aliased 'default' for backwards compatibility
 *********************************************/
.reveal .slides section[data-transition=default].stack,
.reveal.default .slides section.stack {
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal .slides > section[data-transition=default].past,
.reveal .slides > section[data-transition~=default-out].past,
.reveal.default .slides > section:not([data-transition]).past {
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0); }

.reveal .slides > section[data-transition=default].future,
.reveal .slides > section[data-transition~=default-in].future,
.reveal.default .slides > section:not([data-transition]).future {
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0); }

.reveal .slides > section > section[data-transition=default].past,
.reveal .slides > section > section[data-transition~=default-out].past,
.reveal.default .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0);
          transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0); }

.reveal .slides > section > section[data-transition=default].future,
.reveal .slides > section > section[data-transition~=default-in].future,
.reveal.default .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0);
          transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0); }

.reveal .slides section[data-transition=convex].stack,
.reveal.convex .slides section.stack {
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal .slides > section[data-transition=convex].past,
.reveal .slides > section[data-transition~=convex-out].past,
.reveal.convex .slides > section:not([data-transition]).past {
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0); }

.reveal .slides > section[data-transition=convex].future,
.reveal .slides > section[data-transition~=convex-in].future,
.reveal.convex .slides > section:not([data-transition]).future {
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0); }

.reveal .slides > section > section[data-transition=convex].past,
.reveal .slides > section > section[data-transition~=convex-out].past,
.reveal.convex .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0);
          transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0); }

.reveal .slides > section > section[data-transition=convex].future,
.reveal .slides > section > section[data-transition~=convex-in].future,
.reveal.convex .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0);
          transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0); }

/*********************************************
 * CONCAVE TRANSITION
 *********************************************/
.reveal .slides section[data-transition=concave].stack,
.reveal.concave .slides section.stack {
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal .slides > section[data-transition=concave].past,
.reveal .slides > section[data-transition~=concave-out].past,
.reveal.concave .slides > section:not([data-transition]).past {
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0); }

.reveal .slides > section[data-transition=concave].future,
.reveal .slides > section[data-transition~=concave-in].future,
.reveal.concave .slides > section:not([data-transition]).future {
  -webkit-transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0); }

.reveal .slides > section > section[data-transition=concave].past,
.reveal .slides > section > section[data-transition~=concave-out].past,
.reveal.concave .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate3d(0, -80%, 0) rotateX(-70deg) translate3d(0, -80%, 0);
          transform: translate3d(0, -80%, 0) rotateX(-70deg) translate3d(0, -80%, 0); }

.reveal .slides > section > section[data-transition=concave].future,
.reveal .slides > section > section[data-transition~=concave-in].future,
.reveal.concave .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate3d(0, 80%, 0) rotateX(70deg) translate3d(0, 80%, 0);
          transform: translate3d(0, 80%, 0) rotateX(70deg) translate3d(0, 80%, 0); }

/*********************************************
 * ZOOM TRANSITION
 *********************************************/
.reveal .slides section[data-transition=zoom],
.reveal.zoom .slides section:not([data-transition]) {
  transition-timing-function: ease; }

.reveal .slides > section[data-transition=zoom].past,
.reveal .slides > section[data-transition~=zoom-out].past,
.reveal.zoom .slides > section:not([data-transition]).past {
  visibility: hidden;
  -webkit-transform: scale(16);
          transform: scale(16); }

.reveal .slides > section[data-transition=zoom].future,
.reveal .slides > section[data-transition~=zoom-in].future,
.reveal.zoom .slides > section:not([data-transition]).future {
  visibility: hidden;
  -webkit-transform: scale(0.2);
          transform: scale(0.2); }

.reveal .slides > section > section[data-transition=zoom].past,
.reveal .slides > section > section[data-transition~=zoom-out].past,
.reveal.zoom .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate(0, -150%);
          transform: translate(0, -150%); }

.reveal .slides > section > section[data-transition=zoom].future,
.reveal .slides > section > section[data-transition~=zoom-in].future,
.reveal.zoom .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate(0, 150%);
          transform: translate(0, 150%); }

/*********************************************
 * CUBE TRANSITION
 *
 * WARNING:
 * this is deprecated and will be removed in a
 * future version.
 *********************************************/
.reveal.cube .slides {
  -webkit-perspective: 1300px;
          perspective: 1300px; }

.reveal.cube .slides section {
  padding: 30px;
  min-height: 700px;
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden;
  box-sizing: border-box;
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal.center.cube .slides section {
  min-height: 0; }

.reveal.cube .slides section:not(.stack):before {
  content: '';
  position: absolute;
  display: block;
  width: 100%;
  height: 100%;
  left: 0;
  top: 0;
  background: rgba(0, 0, 0, 0.1);
  border-radius: 4px;
  -webkit-transform: translateZ(-20px);
          transform: translateZ(-20px); }

.reveal.cube .slides section:not(.stack):after {
  content: '';
  position: absolute;
  display: block;
  width: 90%;
  height: 30px;
  left: 5%;
  bottom: 0;
  background: none;
  z-index: 1;
  border-radius: 4px;
  box-shadow: 0px 95px 25px rgba(0, 0, 0, 0.2);
  -webkit-transform: translateZ(-90px) rotateX(65deg);
          transform: translateZ(-90px) rotateX(65deg); }

.reveal.cube .slides > section.stack {
  padding: 0;
  background: none; }

.reveal.cube .slides > section.past {
  -webkit-transform-origin: 100% 0%;
          transform-origin: 100% 0%;
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg); }

.reveal.cube .slides > section.future {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg);
          transform: translate3d(100%, 0, 0) rotateY(90deg); }

.reveal.cube .slides > section > section.past {
  -webkit-transform-origin: 0% 100%;
          transform-origin: 0% 100%;
  -webkit-transform: translate3d(0, -100%, 0) rotateX(90deg);
          transform: translate3d(0, -100%, 0) rotateX(90deg); }

.reveal.cube .slides > section > section.future {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(0, 100%, 0) rotateX(-90deg);
          transform: translate3d(0, 100%, 0) rotateX(-90deg); }

/*********************************************
 * PAGE TRANSITION
 *
 * WARNING:
 * this is deprecated and will be removed in a
 * future version.
 *********************************************/
.reveal.page .slides {
  -webkit-perspective-origin: 0% 50%;
          perspective-origin: 0% 50%;
  -webkit-perspective: 3000px;
          perspective: 3000px; }

.reveal.page .slides section {
  padding: 30px;
  min-height: 700px;
  box-sizing: border-box;
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal.page .slides section.past {
  z-index: 12; }

.reveal.page .slides section:not(.stack):before {
  content: '';
  position: absolute;
  display: block;
  width: 100%;
  height: 100%;
  left: 0;
  top: 0;
  background: rgba(0, 0, 0, 0.1);
  -webkit-transform: translateZ(-20px);
          transform: translateZ(-20px); }

.reveal.page .slides section:not(.stack):after {
  content: '';
  position: absolute;
  display: block;
  width: 90%;
  height: 30px;
  left: 5%;
  bottom: 0;
  background: none;
  z-index: 1;
  border-radius: 4px;
  box-shadow: 0px 95px 25px rgba(0, 0, 0, 0.2);
  -webkit-transform: translateZ(-90px) rotateX(65deg); }

.reveal.page .slides > section.stack {
  padding: 0;
  background: none; }

.reveal.page .slides > section.past {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(-40%, 0, 0) rotateY(-80deg);
          transform: translate3d(-40%, 0, 0) rotateY(-80deg); }

.reveal.page .slides > section.future {
  -webkit-transform-origin: 100% 0%;
          transform-origin: 100% 0%;
  -webkit-transform: translate3d(0, 0, 0);
          transform: translate3d(0, 0, 0); }

.reveal.page .slides > section > section.past {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(0, -40%, 0) rotateX(80deg);
          transform: translate3d(0, -40%, 0) rotateX(80deg); }

.reveal.page .slides > section > section.future {
  -webkit-transform-origin: 0% 100%;
          transform-origin: 0% 100%;
  -webkit-transform: translate3d(0, 0, 0);
          transform: translate3d(0, 0, 0); }

/*********************************************
 * FADE TRANSITION
 *********************************************/
.reveal .slides section[data-transition=fade],
.reveal.fade .slides section:not([data-transition]),
.reveal.fade .slides > section > section:not([data-transition]) {
  -webkit-transform: none;
          transform: none;
  transition: opacity 0.5s; }

.reveal.fade.overview .slides section,
.reveal.fade.overview .slides > section > section {
  transition: none; }

/*********************************************
 * NO TRANSITION
 *********************************************/
.reveal .slides section[data-transition=none],
.reveal.none .slides section:not([data-transition]) {
  -webkit-transform: none;
          transform: none;
  transition: none; }

/*********************************************
 * PAUSED MODE
 *********************************************/
.reveal .pause-overlay {
  position: absolute;
  top: 0;
  left: 0;
  width: 100%;
  height: 100%;
  background: black;
  visibility: hidden;
  opacity: 0;
  z-index: 100;
  transition: all 1s ease; }

.reveal .pause-overlay .resume-button {
  position: absolute;
  bottom: 20px;
  right: 20px;
  color: #ccc;
  border-radius: 2px;
  padding: 6px 14px;
  border: 2px solid #ccc;
  font-size: 16px;
  background: transparent;
  cursor: pointer; }
  .reveal .pause-overlay .resume-button:hover {
    color: #fff;
    border-color: #fff; }

.reveal.paused .pause-overlay {
  visibility: visible;
  opacity: 1; }

/*********************************************
 * FALLBACK
 *********************************************/
.no-transforms {
  overflow-y: auto; }

.no-transforms .reveal .slides {
  position: relative;
  width: 80%;
  height: auto !important;
  top: 0;
  left: 50%;
  margin: 0;
  text-align: center; }

.no-transforms .reveal .controls,
.no-transforms .reveal .progress {
  display: none !important; }

.no-transforms .reveal .slides section {
  display: block !important;
  opacity: 1 !important;
  position: relative !important;
  height: auto;
  min-height: 0;
  top: 0;
  left: -50%;
  margin: 70px 0;
  -webkit-transform: none;
          transform: none; }

.no-transforms .reveal .slides section section {
  left: 0; }

.reveal .no-transition,
.reveal .no-transition * {
  transition: none !important; }

/*********************************************
 * PER-SLIDE BACKGROUNDS
 *********************************************/
.reveal .backgrounds {
  position: absolute;
  width: 100%;
  height: 100%;
  top: 0;
  left: 0;
  -webkit-perspective: 600px;
          perspective: 600px; }

.reveal .slide-background {
  display: none;
  position: absolute;
  width: 100%;
  height: 100%;
  opacity: 0;
  visibility: hidden;
  overflow: hidden;
  background-color: transparent;
  transition: all 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985); }

.reveal .slide-background-content {
  position: absolute;
  width: 100%;
  height: 100%;
  background-position: 50% 50%;
  background-repeat: no-repeat;
  background-size: cover; }

.reveal .slide-background.stack {
  display: block; }

.reveal .slide-background.present {
  opacity: 1;
  visibility: visible;
  z-index: 2; }

.print-pdf .reveal .slide-background {
  opacity: 1 !important;
  visibility: visible !important; }

/* Video backgrounds */
.reveal .slide-background video {
  position: absolute;
  width: 100%;
  height: 100%;
  max-width: none;
  max-height: none;
  top: 0;
  left: 0;
  -o-object-fit: cover;
     object-fit: cover; }

.reveal .slide-background[data-background-size="contain"] video {
  -o-object-fit: contain;
     object-fit: contain; }

/* Immediate transition style */
.reveal[data-background-transition=none] > .backgrounds .slide-background,
.reveal > .backgrounds .slide-background[data-background-transition=none] {
  transition: none; }

/* Slide */
.reveal[data-background-transition=slide] > .backgrounds .slide-background,
.reveal > .backgrounds .slide-background[data-background-transition=slide] {
  opacity: 1;
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden; }

.reveal[data-background-transition=slide] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=slide] {
  -webkit-transform: translate(-100%, 0);
          transform: translate(-100%, 0); }

.reveal[data-background-transition=slide] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=slide] {
  -webkit-transform: translate(100%, 0);
          transform: translate(100%, 0); }

.reveal[data-background-transition=slide] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=slide] {
  -webkit-transform: translate(0, -100%);
          transform: translate(0, -100%); }

.reveal[data-background-transition=slide] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=slide] {
  -webkit-transform: translate(0, 100%);
          transform: translate(0, 100%); }

/* Convex */
.reveal[data-background-transition=convex] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0); }

.reveal[data-background-transition=convex] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0); }

.reveal[data-background-transition=convex] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(0, -100%, 0) rotateX(90deg) translate3d(0, -100%, 0);
          transform: translate3d(0, -100%, 0) rotateX(90deg) translate3d(0, -100%, 0); }

.reveal[data-background-transition=convex] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(0, 100%, 0) rotateX(-90deg) translate3d(0, 100%, 0);
          transform: translate3d(0, 100%, 0) rotateX(-90deg) translate3d(0, 100%, 0); }

/* Concave */
.reveal[data-background-transition=concave] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0); }

.reveal[data-background-transition=concave] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0); }

.reveal[data-background-transition=concave] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(0, -100%, 0) rotateX(-90deg) translate3d(0, -100%, 0);
          transform: translate3d(0, -100%, 0) rotateX(-90deg) translate3d(0, -100%, 0); }

.reveal[data-background-transition=concave] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(0, 100%, 0) rotateX(90deg) translate3d(0, 100%, 0);
          transform: translate3d(0, 100%, 0) rotateX(90deg) translate3d(0, 100%, 0); }

/* Zoom */
.reveal[data-background-transition=zoom] > .backgrounds .slide-background,
.reveal > .backgrounds .slide-background[data-background-transition=zoom] {
  transition-timing-function: ease; }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(16);
          transform: scale(16); }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(0.2);
          transform: scale(0.2); }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(16);
          transform: scale(16); }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(0.2);
          transform: scale(0.2); }

/* Global transition speed settings */
.reveal[data-transition-speed="fast"] > .backgrounds .slide-background {
  transition-duration: 400ms; }

.reveal[data-transition-speed="slow"] > .backgrounds .slide-background {
  transition-duration: 1200ms; }

/*********************************************
 * OVERVIEW
 *********************************************/
.reveal.overview {
  -webkit-perspective-origin: 50% 50%;
          perspective-origin: 50% 50%;
  -webkit-perspective: 700px;
          perspective: 700px; }
  .reveal.overview .slides {
    -moz-transform-style: preserve-3d; }
  .reveal.overview .slides section {
    height: 100%;
    top: 0 !important;
    opacity: 1 !important;
    overflow: hidden;
    visibility: visible !important;
    cursor: pointer;
    box-sizing: border-box; }
  .reveal.overview .slides section:hover,
  .reveal.overview .slides section.present {
    outline: 10px solid rgba(150, 150, 150, 0.4);
    outline-offset: 10px; }
  .reveal.overview .slides section .fragment {
    opacity: 1;
    transition: none; }
  .reveal.overview .slides section:after,
  .reveal.overview .slides section:before {
    display: none !important; }
  .reveal.overview .slides > section.stack {
    padding: 0;
    top: 0 !important;
    background: none;
    outline: none;
    overflow: visible; }
  .reveal.overview .backgrounds {
    -webkit-perspective: inherit;
            perspective: inherit;
    -moz-transform-style: preserve-3d; }
  .reveal.overview .backgrounds .slide-background {
    opacity: 1;
    visibility: visible;
    outline: 10px solid rgba(150, 150, 150, 0.1);
    outline-offset: 10px; }
  .reveal.overview .backgrounds .slide-background.stack {
    overflow: visible; }

.reveal.overview .slides section,
.reveal.overview-deactivating .slides section {
  transition: none; }

.reveal.overview .backgrounds .slide-background,
.reveal.overview-deactivating .backgrounds .slide-background {
  transition: none; }

/*********************************************
 * RTL SUPPORT
 *********************************************/
.reveal.rtl .slides,
.reveal.rtl .slides h1,
.reveal.rtl .slides h2,
.reveal.rtl .slides h3,
.reveal.rtl .slides h4,
.reveal.rtl .slides h5,
.reveal.rtl .slides h6 {
  direction: rtl;
  font-family: sans-serif; }

.reveal.rtl pre,
.reveal.rtl code {
  direction: ltr; }

.reveal.rtl ol,
.reveal.rtl ul {
  text-align: right; }

.reveal.rtl .progress span {
  float: right; }

/*********************************************
 * PARALLAX BACKGROUND
 *********************************************/
.reveal.has-parallax-background .backgrounds {
  transition: all 0.8s ease; }

/* Global transition speed settings */
.reveal.has-parallax-background[data-transition-speed="fast"] .backgrounds {
  transition-duration: 400ms; }

.reveal.has-parallax-background[data-transition-speed="slow"] .backgrounds {
  transition-duration: 1200ms; }

/*********************************************
 * LINK PREVIEW OVERLAY
 *********************************************/
.reveal .overlay {
  position: absolute;
  top: 0;
  left: 0;
  width: 100%;
  height: 100%;
  z-index: 1000;
  background: rgba(0, 0, 0, 0.9);
  opacity: 0;
  visibility: hidden;
  transition: all 0.3s ease; }

.reveal .overlay.visible {
  opacity: 1;
  visibility: visible; }

.reveal .overlay .spinner {
  position: absolute;
  display: block;
  top: 50%;
  left: 50%;
  width: 32px;
  height: 32px;
  margin: -16px 0 0 -16px;
  z-index: 10;
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        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
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todayMarker off
dateFormat  YYYY
axisFormat %Y

section &#x795E;&#x7ECF;&#x7F51;&#x7EDC;
&#x6A21;&#x578B;&#x63D0;&#x51FA;: done, 1943, 1969
1943 MP&#x795E;&#x7ECF;&#x7F51;&#x7EDC;: 1943, milestone
1958 Rosenblatt&#x63D0;&#x51FA;&#x611F;&#x77E5;&#x673A;: 1958, milestone
1969 Minsky&#x51FA;&#x7248;&#x300A;&#x611F;&#x77E5;&#x673A;&#x300B;: 1969, milestone
&#x51B0;&#x6CB3;&#x671F;: done, 1969, 1983
1974 &#x53CD;&#x5411;&#x4F20;&#x64AD;&#x88AB;&#x63D0;&#x51FA;: 1974, milestone
1980 &#x5E26;&#x5377;&#x79EF;&#x548C;&#x5B50;&#x91C7;&#x6837;&#x7684;&#x65B0;&#x77E5;&#x673A;: 1980, milestone
&#x590D;&#x5174;: done, 1983, 1995
1983 Hopfield&#x7F51;&#x7EDC;: 1983, milestone
1984 Boltzmann&#x673A;: 1984, milestone
1986 &#x53CD;&#x5411;&#x4F20;&#x64AD;&#x88AB;&#x91CD;&#x65B0;&#x63D0;&#x51FA;: 1986, milestone
&#x4E8C;&#x6B21;&#x51B0;&#x6CB3;: done, 1995, 2006
1995 &#x7EDF;&#x8BA1;&#x673A;&#x5668;&#x5B66;&#x4E60;&#x5174;&#x8D77;: 1995, milestone
&#x6DF1;&#x5EA6;&#x5B66;&#x4E60;: active, 2006, 2020
2012 DNN&#x5F15;&#x8D77;&#x8F70;&#x52A8;: 2012, milestone
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<title>12</title>
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</g>
<!-- 12&#45;&gt;21 -->
<g id="edge2" class="edge">
<title>12-&gt;21</title>
<path fill="none" stroke="#859900" d="M56.7953,-77.9183C61.8997,-82.0393 67.4686,-86.9571 72,-92 98.7765,-121.7986 96.2989,-136.2691 122,-167 124.7142,-170.2454 127.8067,-173.5191 130.8888,-176.58"/>
<polygon fill="#859900" stroke="#859900" points="134.4788,-180.0612 129.3229,-178.1958 132.684,-178.3208 130.8893,-176.5805 130.8893,-176.5805 130.8893,-176.5805 132.684,-178.3208 132.4556,-174.9652 134.4788,-180.0612 134.4788,-180.0612"/>
</g>
<!-- 12&#45;&gt;22 -->
<g id="edge7" class="edge">
<title>12-&gt;22</title>
<path fill="none" stroke="#586e75" d="M56.7388,-77.4284C75.5394,-90.7307 108.0936,-113.7644 128.8123,-128.4238"/>
<polygon fill="#586e75" stroke="#586e75" points="133.0446,-131.4184 127.6634,-130.3671 131.0038,-129.9744 128.963,-128.5304 128.963,-128.5304 128.963,-128.5304 131.0038,-129.9744 130.2626,-126.6936 133.0446,-131.4184 133.0446,-131.4184"/>
</g>
<!-- 12&#45;&gt;23 -->
<g id="edge8" class="edge">
<title>12-&gt;23</title>
<path fill="none" stroke="#586e75" d="M59.9815,-71.2409C77.9064,-75.4685 105.5043,-81.9774 125.0114,-86.5782"/>
<polygon fill="#586e75" stroke="#586e75" points="130.2841,-87.8217 124.9011,-88.8638 127.8509,-87.2478 125.4176,-86.6739 125.4176,-86.6739 125.4176,-86.6739 127.8509,-87.2478 125.9341,-84.484 130.2841,-87.8217 130.2841,-87.8217"/>
</g>
<!-- 12&#45;&gt;24 -->
<g id="edge9" class="edge">
<title>12-&gt;24</title>
<path fill="none" stroke="#586e75" d="M59.9815,-62.7591C77.9064,-58.5315 105.5043,-52.0226 125.0114,-47.4218"/>
<polygon fill="#586e75" stroke="#586e75" points="130.2841,-46.1783 125.9341,-49.516 127.8509,-46.7522 125.4176,-47.3261 125.4176,-47.3261 125.4176,-47.3261 127.8509,-46.7522 124.9011,-45.1362 130.2841,-46.1783 130.2841,-46.1783"/>
</g>
<!-- 13 -->
<g id="node3" class="node">
<title>13</title>
<ellipse fill="none" stroke="#859900" cx="42" cy="-167" rx="18" ry="18"/>
</g>
<!-- 13&#45;&gt;21 -->
<g id="edge3" class="edge">
<title>13-&gt;21</title>
<path fill="none" stroke="#859900" d="M59.9815,-171.2409C77.9064,-175.4685 105.5043,-181.9774 125.0114,-186.5782"/>
<polygon fill="#859900" stroke="#859900" points="130.2841,-187.8217 124.9011,-188.8638 127.8509,-187.2478 125.4176,-186.6739 125.4176,-186.6739 125.4176,-186.6739 127.8509,-187.2478 125.9341,-184.484 130.2841,-187.8217 130.2841,-187.8217"/>
</g>
<!-- 13&#45;&gt;22 -->
<g id="edge10" class="edge">
<title>13-&gt;22</title>
<path fill="none" stroke="#586e75" d="M59.9815,-162.7591C77.9064,-158.5315 105.5043,-152.0226 125.0114,-147.4218"/>
<polygon fill="#586e75" stroke="#586e75" points="130.2841,-146.1783 125.9341,-149.516 127.8509,-146.7522 125.4176,-147.3261 125.4176,-147.3261 125.4176,-147.3261 127.8509,-146.7522 124.9011,-145.1362 130.2841,-146.1783 130.2841,-146.1783"/>
</g>
<!-- 13&#45;&gt;23 -->
<g id="edge11" class="edge">
<title>13-&gt;23</title>
<path fill="none" stroke="#586e75" d="M56.7388,-156.5716C75.5394,-143.2693 108.0936,-120.2356 128.8123,-105.5762"/>
<polygon fill="#586e75" stroke="#586e75" points="133.0446,-102.5816 130.2626,-107.3064 131.0038,-104.0256 128.963,-105.4696 128.963,-105.4696 128.963,-105.4696 131.0038,-104.0256 127.6634,-103.6329 133.0446,-102.5816 133.0446,-102.5816"/>
</g>
<!-- 13&#45;&gt;24 -->
<g id="edge12" class="edge">
<title>13-&gt;24</title>
<path fill="none" stroke="#586e75" d="M56.7953,-156.0817C61.8997,-151.9607 67.4686,-147.0429 72,-142 98.7765,-112.2014 96.2989,-97.7309 122,-67 124.7142,-63.7546 127.8067,-60.4809 130.8888,-57.42"/>
<polygon fill="#586e75" stroke="#586e75" points="134.4788,-53.9388 132.4556,-59.0348 132.684,-55.6792 130.8893,-57.4195 130.8893,-57.4195 130.8893,-57.4195 132.684,-55.6792 129.3229,-55.8042 134.4788,-53.9388 134.4788,-53.9388"/>
</g>
<!-- 31 -->
<g id="node8" class="node">
<title>31</title>
<ellipse fill="none" stroke="#586e75" cx="242" cy="-42" rx="18" ry="18"/>
</g>
<!-- 21&#45;&gt;31 -->
<g id="edge13" class="edge">
<title>21-&gt;31</title>
<path fill="none" stroke="#586e75" d="M158.9051,-177.3616C161.306,-174.0007 163.7902,-170.4115 166,-167 193.9332,-123.8776 196.0668,-110.1224 224,-67 225.3121,-64.9744 226.7209,-62.8862 228.1515,-60.8203"/>
<polygon fill="#586e75" stroke="#586e75" points="231.0949,-56.6384 230.057,-62.0223 229.6559,-58.6828 228.217,-60.7272 228.217,-60.7272 228.217,-60.7272 229.6559,-58.6828 226.3771,-59.4322 231.0949,-56.6384 231.0949,-56.6384"/>
</g>
<!-- 32 -->
<g id="node9" class="node">
<title>32</title>
<ellipse fill="none" stroke="#586e75" cx="242" cy="-92" rx="18" ry="18"/>
</g>
<!-- 21&#45;&gt;32 -->
<g id="edge14" class="edge">
<title>21-&gt;32</title>
<path fill="none" stroke="#586e75" d="M160.6773,-178.5136C177.5905,-160.5208 207.5422,-128.6573 225.9939,-109.0278"/>
<polygon fill="#586e75" stroke="#586e75" points="229.5236,-105.2728 227.7384,-110.457 227.8113,-107.0944 226.099,-108.9159 226.099,-108.9159 226.099,-108.9159 227.8113,-107.0944 224.4596,-107.3749 229.5236,-105.2728 229.5236,-105.2728"/>
</g>
<!-- 33 -->
<g id="node10" class="node">
<title>33</title>
<ellipse fill="none" stroke="#586e75" cx="242" cy="-142" rx="18" ry="18"/>
</g>
<!-- 21&#45;&gt;33 -->
<g id="edge15" class="edge">
<title>21-&gt;33</title>
<path fill="none" stroke="#586e75" d="M163.9458,-183.5182C179.7458,-175.1139 204.0206,-162.2018 221.3005,-153.0104"/>
<polygon fill="#586e75" stroke="#586e75" points="225.976,-150.5234 222.6182,-154.858 223.7688,-151.6975 221.5616,-152.8715 221.5616,-152.8715 221.5616,-152.8715 223.7688,-151.6975 220.5049,-150.8851 225.976,-150.5234 225.976,-150.5234"/>
</g>
<!-- 34 -->
<g id="node11" class="node">
<title>34</title>
<ellipse fill="none" stroke="#586e75" cx="242" cy="-192" rx="18" ry="18"/>
</g>
<!-- 21&#45;&gt;34 -->
<g id="edge16" class="edge">
<title>21-&gt;34</title>
<path fill="none" stroke="#586e75" d="M166.1241,-192C181.0789,-192 202.4017,-192 218.6371,-192"/>
<polygon fill="#586e75" stroke="#586e75" points="223.9315,-192 218.9315,-194.2501 221.4315,-192 218.9315,-192.0001 218.9315,-192.0001 218.9315,-192.0001 221.4315,-192 218.9315,-189.7501 223.9315,-192 223.9315,-192"/>
</g>
<!-- 22&#45;&gt;31 -->
<g id="edge17" class="edge">
<title>22-&gt;31</title>
<path fill="none" stroke="#586e75" d="M160.6773,-128.5136C177.5905,-110.5208 207.5422,-78.6573 225.9939,-59.0278"/>
<polygon fill="#586e75" stroke="#586e75" points="229.5236,-55.2728 227.7384,-60.457 227.8113,-57.0944 226.099,-58.9159 226.099,-58.9159 226.099,-58.9159 227.8113,-57.0944 224.4596,-57.3749 229.5236,-55.2728 229.5236,-55.2728"/>
</g>
<!-- 22&#45;&gt;32 -->
<g id="edge18" class="edge">
<title>22-&gt;32</title>
<path fill="none" stroke="#586e75" d="M163.9458,-133.5182C179.7458,-125.1139 204.0206,-112.2018 221.3005,-103.0104"/>
<polygon fill="#586e75" stroke="#586e75" points="225.976,-100.5234 222.6182,-104.858 223.7688,-101.6975 221.5616,-102.8715 221.5616,-102.8715 221.5616,-102.8715 223.7688,-101.6975 220.5049,-100.8851 225.976,-100.5234 225.976,-100.5234"/>
</g>
<!-- 22&#45;&gt;33 -->
<g id="edge19" class="edge">
<title>22-&gt;33</title>
<path fill="none" stroke="#586e75" d="M166.1241,-142C181.0789,-142 202.4017,-142 218.6371,-142"/>
<polygon fill="#586e75" stroke="#586e75" points="223.9315,-142 218.9315,-144.2501 221.4315,-142 218.9315,-142.0001 218.9315,-142.0001 218.9315,-142.0001 221.4315,-142 218.9315,-139.7501 223.9315,-142 223.9315,-142"/>
</g>
<!-- 22&#45;&gt;34 -->
<g id="edge20" class="edge">
<title>22-&gt;34</title>
<path fill="none" stroke="#586e75" d="M163.9458,-150.4818C179.7458,-158.8861 204.0206,-171.7982 221.3005,-180.9896"/>
<polygon fill="#586e75" stroke="#586e75" points="225.976,-183.4766 220.5049,-183.1149 223.7688,-182.3025 221.5616,-181.1285 221.5616,-181.1285 221.5616,-181.1285 223.7688,-182.3025 222.6182,-179.142 225.976,-183.4766 225.976,-183.4766"/>
</g>
<!-- 23&#45;&gt;31 -->
<g id="edge21" class="edge">
<title>23-&gt;31</title>
<path fill="none" stroke="#586e75" d="M163.9458,-83.5182C179.7458,-75.1139 204.0206,-62.2018 221.3005,-53.0104"/>
<polygon fill="#586e75" stroke="#586e75" points="225.976,-50.5234 222.6182,-54.858 223.7688,-51.6975 221.5616,-52.8715 221.5616,-52.8715 221.5616,-52.8715 223.7688,-51.6975 220.5049,-50.8851 225.976,-50.5234 225.976,-50.5234"/>
</g>
<!-- 23&#45;&gt;32 -->
<g id="edge22" class="edge">
<title>23-&gt;32</title>
<path fill="none" stroke="#586e75" d="M166.1241,-92C181.0789,-92 202.4017,-92 218.6371,-92"/>
<polygon fill="#586e75" stroke="#586e75" points="223.9315,-92 218.9315,-94.2501 221.4315,-92 218.9315,-92.0001 218.9315,-92.0001 218.9315,-92.0001 221.4315,-92 218.9315,-89.7501 223.9315,-92 223.9315,-92"/>
</g>
<!-- 23&#45;&gt;33 -->
<g id="edge23" class="edge">
<title>23-&gt;33</title>
<path fill="none" stroke="#586e75" d="M163.9458,-100.4818C179.7458,-108.8861 204.0206,-121.7982 221.3005,-130.9896"/>
<polygon fill="#586e75" stroke="#586e75" points="225.976,-133.4766 220.5049,-133.1149 223.7688,-132.3025 221.5616,-131.1285 221.5616,-131.1285 221.5616,-131.1285 223.7688,-132.3025 222.6182,-129.142 225.976,-133.4766 225.976,-133.4766"/>
</g>
<!-- 23&#45;&gt;34 -->
<g id="edge24" class="edge">
<title>23-&gt;34</title>
<path fill="none" stroke="#586e75" d="M160.6773,-105.4864C177.5905,-123.4792 207.5422,-155.3427 225.9939,-174.9722"/>
<polygon fill="#586e75" stroke="#586e75" points="229.5236,-178.7272 224.4596,-176.6251 227.8113,-176.9056 226.099,-175.0841 226.099,-175.0841 226.099,-175.0841 227.8113,-176.9056 227.7384,-173.543 229.5236,-178.7272 229.5236,-178.7272"/>
</g>
<!-- 24&#45;&gt;31 -->
<g id="edge25" class="edge">
<title>24-&gt;31</title>
<path fill="none" stroke="#586e75" d="M166.1241,-42C181.0789,-42 202.4017,-42 218.6371,-42"/>
<polygon fill="#586e75" stroke="#586e75" points="223.9315,-42 218.9315,-44.2501 221.4315,-42 218.9315,-42.0001 218.9315,-42.0001 218.9315,-42.0001 221.4315,-42 218.9315,-39.7501 223.9315,-42 223.9315,-42"/>
</g>
<!-- 24&#45;&gt;32 -->
<g id="edge26" class="edge">
<title>24-&gt;32</title>
<path fill="none" stroke="#586e75" d="M163.9458,-50.4818C179.7458,-58.8861 204.0206,-71.7982 221.3005,-80.9896"/>
<polygon fill="#586e75" stroke="#586e75" points="225.976,-83.4766 220.5049,-83.1149 223.7688,-82.3025 221.5616,-81.1285 221.5616,-81.1285 221.5616,-81.1285 223.7688,-82.3025 222.6182,-79.142 225.976,-83.4766 225.976,-83.4766"/>
</g>
<!-- 24&#45;&gt;33 -->
<g id="edge27" class="edge">
<title>24-&gt;33</title>
<path fill="none" stroke="#586e75" d="M160.6773,-55.4864C177.5905,-73.4792 207.5422,-105.3427 225.9939,-124.9722"/>
<polygon fill="#586e75" stroke="#586e75" points="229.5236,-128.7272 224.4596,-126.6251 227.8113,-126.9056 226.099,-125.0841 226.099,-125.0841 226.099,-125.0841 227.8113,-126.9056 227.7384,-123.543 229.5236,-128.7272 229.5236,-128.7272"/>
</g>
<!-- 24&#45;&gt;34 -->
<g id="edge28" class="edge">
<title>24-&gt;34</title>
<path fill="none" stroke="#586e75" d="M158.9051,-56.6384C161.306,-59.9993 163.7902,-63.5885 166,-67 193.9332,-110.1224 196.0668,-123.8776 224,-167 225.3121,-169.0256 226.7209,-171.1138 228.1515,-173.1797"/>
<polygon fill="#586e75" stroke="#586e75" points="231.0949,-177.3616 226.3771,-174.5678 229.6559,-175.3172 228.217,-173.2728 228.217,-173.2728 228.217,-173.2728 229.6559,-175.3172 230.057,-171.9777 231.0949,-177.3616 231.0949,-177.3616"/>
</g>
<!-- 41 -->
<g id="node12" class="node">
<title>41</title>
<ellipse fill="none" stroke="#586e75" cx="336" cy="-42" rx="18" ry="18"/>
</g>
<!-- 31&#45;&gt;41 -->
<g id="edge29" class="edge">
<title>31-&gt;41</title>
<path fill="none" stroke="#586e75" d="M260.1241,-42C275.0789,-42 296.4017,-42 312.6371,-42"/>
<polygon fill="#586e75" stroke="#586e75" points="317.9315,-42 312.9315,-44.2501 315.4315,-42 312.9315,-42.0001 312.9315,-42.0001 312.9315,-42.0001 315.4315,-42 312.9315,-39.7501 317.9315,-42 317.9315,-42"/>
</g>
<!-- 42 -->
<g id="node13" class="node">
<title>42</title>
<ellipse fill="none" stroke="#586e75" cx="336" cy="-92" rx="18" ry="18"/>
</g>
<!-- 31&#45;&gt;42 -->
<g id="edge30" class="edge">
<title>31-&gt;42</title>
<path fill="none" stroke="#586e75" d="M257.9458,-50.4818C273.7458,-58.8861 298.0206,-71.7982 315.3005,-80.9896"/>
<polygon fill="#586e75" stroke="#586e75" points="319.976,-83.4766 314.5049,-83.1149 317.7688,-82.3025 315.5616,-81.1285 315.5616,-81.1285 315.5616,-81.1285 317.7688,-82.3025 316.6182,-79.142 319.976,-83.4766 319.976,-83.4766"/>
</g>
<!-- 43 -->
<g id="node14" class="node">
<title>43</title>
<ellipse fill="none" stroke="#586e75" cx="336" cy="-142" rx="18" ry="18"/>
</g>
<!-- 31&#45;&gt;43 -->
<g id="edge31" class="edge">
<title>31-&gt;43</title>
<path fill="none" stroke="#586e75" d="M254.6773,-55.4864C271.5905,-73.4792 301.5422,-105.3427 319.9939,-124.9722"/>
<polygon fill="#586e75" stroke="#586e75" points="323.5236,-128.7272 318.4596,-126.6251 321.8113,-126.9056 320.099,-125.0841 320.099,-125.0841 320.099,-125.0841 321.8113,-126.9056 321.7384,-123.543 323.5236,-128.7272 323.5236,-128.7272"/>
</g>
<!-- 44 -->
<g id="node15" class="node">
<title>44</title>
<ellipse fill="none" stroke="#586e75" cx="336" cy="-192" rx="18" ry="18"/>
</g>
<!-- 31&#45;&gt;44 -->
<g id="edge32" class="edge">
<title>31-&gt;44</title>
<path fill="none" stroke="#586e75" d="M252.9051,-56.6384C255.306,-59.9993 257.7902,-63.5885 260,-67 287.9332,-110.1224 290.0668,-123.8776 318,-167 319.3121,-169.0256 320.7209,-171.1138 322.1515,-173.1797"/>
<polygon fill="#586e75" stroke="#586e75" points="325.0949,-177.3616 320.3771,-174.5678 323.6559,-175.3172 322.217,-173.2728 322.217,-173.2728 322.217,-173.2728 323.6559,-175.3172 324.057,-171.9777 325.0949,-177.3616 325.0949,-177.3616"/>
</g>
<!-- 32&#45;&gt;41 -->
<g id="edge33" class="edge">
<title>32-&gt;41</title>
<path fill="none" stroke="#586e75" d="M257.9458,-83.5182C273.7458,-75.1139 298.0206,-62.2018 315.3005,-53.0104"/>
<polygon fill="#586e75" stroke="#586e75" points="319.976,-50.5234 316.6182,-54.858 317.7688,-51.6975 315.5616,-52.8715 315.5616,-52.8715 315.5616,-52.8715 317.7688,-51.6975 314.5049,-50.8851 319.976,-50.5234 319.976,-50.5234"/>
</g>
<!-- 32&#45;&gt;42 -->
<g id="edge34" class="edge">
<title>32-&gt;42</title>
<path fill="none" stroke="#586e75" d="M260.1241,-92C275.0789,-92 296.4017,-92 312.6371,-92"/>
<polygon fill="#586e75" stroke="#586e75" points="317.9315,-92 312.9315,-94.2501 315.4315,-92 312.9315,-92.0001 312.9315,-92.0001 312.9315,-92.0001 315.4315,-92 312.9315,-89.7501 317.9315,-92 317.9315,-92"/>
</g>
<!-- 32&#45;&gt;43 -->
<g id="edge35" class="edge">
<title>32-&gt;43</title>
<path fill="none" stroke="#586e75" d="M257.9458,-100.4818C273.7458,-108.8861 298.0206,-121.7982 315.3005,-130.9896"/>
<polygon fill="#586e75" stroke="#586e75" points="319.976,-133.4766 314.5049,-133.1149 317.7688,-132.3025 315.5616,-131.1285 315.5616,-131.1285 315.5616,-131.1285 317.7688,-132.3025 316.6182,-129.142 319.976,-133.4766 319.976,-133.4766"/>
</g>
<!-- 32&#45;&gt;44 -->
<g id="edge36" class="edge">
<title>32-&gt;44</title>
<path fill="none" stroke="#586e75" d="M254.6773,-105.4864C271.5905,-123.4792 301.5422,-155.3427 319.9939,-174.9722"/>
<polygon fill="#586e75" stroke="#586e75" points="323.5236,-178.7272 318.4596,-176.6251 321.8113,-176.9056 320.099,-175.0841 320.099,-175.0841 320.099,-175.0841 321.8113,-176.9056 321.7384,-173.543 323.5236,-178.7272 323.5236,-178.7272"/>
</g>
<!-- 33&#45;&gt;41 -->
<g id="edge37" class="edge">
<title>33-&gt;41</title>
<path fill="none" stroke="#586e75" d="M254.6773,-128.5136C271.5905,-110.5208 301.5422,-78.6573 319.9939,-59.0278"/>
<polygon fill="#586e75" stroke="#586e75" points="323.5236,-55.2728 321.7384,-60.457 321.8113,-57.0944 320.099,-58.9159 320.099,-58.9159 320.099,-58.9159 321.8113,-57.0944 318.4596,-57.3749 323.5236,-55.2728 323.5236,-55.2728"/>
</g>
<!-- 33&#45;&gt;42 -->
<g id="edge38" class="edge">
<title>33-&gt;42</title>
<path fill="none" stroke="#586e75" d="M257.9458,-133.5182C273.7458,-125.1139 298.0206,-112.2018 315.3005,-103.0104"/>
<polygon fill="#586e75" stroke="#586e75" points="319.976,-100.5234 316.6182,-104.858 317.7688,-101.6975 315.5616,-102.8715 315.5616,-102.8715 315.5616,-102.8715 317.7688,-101.6975 314.5049,-100.8851 319.976,-100.5234 319.976,-100.5234"/>
</g>
<!-- 33&#45;&gt;43 -->
<g id="edge39" class="edge">
<title>33-&gt;43</title>
<path fill="none" stroke="#586e75" d="M260.1241,-142C275.0789,-142 296.4017,-142 312.6371,-142"/>
<polygon fill="#586e75" stroke="#586e75" points="317.9315,-142 312.9315,-144.2501 315.4315,-142 312.9315,-142.0001 312.9315,-142.0001 312.9315,-142.0001 315.4315,-142 312.9315,-139.7501 317.9315,-142 317.9315,-142"/>
</g>
<!-- 33&#45;&gt;44 -->
<g id="edge40" class="edge">
<title>33-&gt;44</title>
<path fill="none" stroke="#586e75" d="M257.9458,-150.4818C273.7458,-158.8861 298.0206,-171.7982 315.3005,-180.9896"/>
<polygon fill="#586e75" stroke="#586e75" points="319.976,-183.4766 314.5049,-183.1149 317.7688,-182.3025 315.5616,-181.1285 315.5616,-181.1285 315.5616,-181.1285 317.7688,-182.3025 316.6182,-179.142 319.976,-183.4766 319.976,-183.4766"/>
</g>
<!-- 34&#45;&gt;41 -->
<g id="edge41" class="edge">
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</p><p>&#x53EA;&#x8981;&#x5B58;&#x5728;&#x9690;&#x85CF;&#x5C42;&#xFF0C;&#x6A21;&#x578B;&#x5C31;&#x62E5;&#x6709;&#x4E86;&#x975E;&#x7EBF;&#x6027;&#x9884;&#x6D4B;&#x80FD;&#x529B;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="286" class="slide " data-line="286" data-h="1" data-v="4">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x5F62;&#x5F0F;&#x5316;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<ul>
<li><span class="mathjax-exps">$L$</span>&#xFF1A;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7684;&#x5C42;&#x6570;</li>
<li><span class="mathjax-exps">$n_l$</span>&#xFF1A;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#x795E;&#x7ECF;&#x5143;&#x7684;&#x4E2A;&#x6570;</li>
<li><span class="mathjax-exps">$h_l(\cdot)$</span>&#xFF1A;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#x7684;&#x6FC0;&#x6D3B;&#x51FD;&#x6570;</li>
<li><span class="mathjax-exps">$\Wv_l \in \Rbb^{n_l \times n_{l-1}}$</span>&#xFF1A;&#x7B2C;<span class="mathjax-exps">$l-1$</span>&#x5C42;&#x5230;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#x7684;&#x6743;&#x91CD;&#x77E9;&#x9635;</li>
<li><span class="mathjax-exps">$\bv_l \in \Rbb^{n_l}$</span>&#xFF1A;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#x7684;&#x622A;&#x8DDD;</li>
<li><span class="mathjax-exps">$\zv_l \in \Rbb^{n_l}$</span>&#xFF1A;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#x795E;&#x7ECF;&#x5143;&#x7684;&#x8F93;&#x5165;</li>
<li><span class="mathjax-exps">$\av_l \in \Rbb^{n_l}$</span>&#xFF1A;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#x795E;&#x7ECF;&#x5143;&#x7684;&#x8F93;&#x51FA;</li>
</ul>
<br>
<div>
    $$
        \begin{align*}
            \zv_l &amp; = \Wv_l ~ \av_{l-1} + \bv_l \\
            \av_l &amp; = h_l (\zv_l) \\
            \xv &amp; = \av_0 \xrightarrow{\Wv_1,\bv_1} \zv_1 \xrightarrow{h_1} \av_1 \xrightarrow{\Wv_2,\bv_2} \cdots \xrightarrow{\Wv_L,\bv_L} \zv_L \xrightarrow{h_L} \av_L = \hat{\yv}
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="324" class="slide " data-line="324" data-h="2" data-v="0">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6FC0;&#x6D3B;&#x51FD;&#x6570;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x6700;&#x65E9;&#x7684; MP &#x6A21;&#x578B;&#x91C7;&#x7528;&#x9636;&#x8DC3;&#x51FD;&#x6570;<span class="mathjax-exps">$\sgn(z) = 1_{z \geq 0}$</span>&#x4F5C;&#x4E3A;&#x6FC0;&#x6D3B;&#x51FD;&#x6570;</p>
<br>
<p>&#x6539;&#x8FDB;&#xFF1A;</p>
<ul>
<li>&#x8FDE;&#x7EED;&#x5E76;&#x51E0;&#x4E4E;&#x5904;&#x5904;&#x53EF;&#x5BFC;&#xFF0C;&#x53EF;&#x4EE5;&#x9AD8;&#x6548;&#x8BA1;&#x7B97;</li>
<li>&#x5BFC;&#x6570;&#x7684;&#x503C;&#x57DF;&#x5728;&#x5408;&#x9002;&#x7684;&#x8303;&#x56F4;&#x5185;&#xFF0C;&#x5426;&#x5219;&#x5F71;&#x54CD;&#x7528;&#x68AF;&#x5EA6;&#x4E0B;&#x964D;&#x8FDB;&#x884C;&#x8BAD;&#x7EC3;</li>
</ul>
<br>
<p>&#x5E38;&#x89C1;&#x7684;&#x6709;</p>
<ul>
<li>Sigmoid &#x578B;&#xFF1A;Logistic &#x51FD;&#x6570;&#xFF0C;Tanh &#x51FD;&#x6570;</li>
<li>ReLU&#xFF0C;&#x5E26;&#x6CC4;&#x6F0F;&#x7684; ReLU&#xFF0C;&#x5E26;&#x53C2;&#x6570;&#x7684; ReLU&#xFF0C;ELU&#xFF0C;Softplus</li>
<li>Swish &#x51FD;&#x6570;</li>
<li>Maxout &#x5355;&#x5143;</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="360" class="slide " data-line="360" data-h="2" data-v="1">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">Sigmoid&#x578B;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
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" style="margin-top:2rem" height="420px">
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section><section vertical="true" data-notes lineno="380" class="slide " data-line="380" data-h="2" data-v="2">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">Logistic&#x51FD;&#x6570;</h5>
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<p>&#x5C06;<span class="mathjax-exps">$\Rbb$</span>&#x201C;&#x6324;&#x538B;&#x201D;&#x5230;<span class="mathjax-exps">$[0,1]$</span>&#xFF0C;&#x8F93;&#x51FA;&#x62E5;&#x6709;&#x6982;&#x7387;&#x610F;&#x4E49;&#xFF1A;</p>
<div>
    $$
        \begin{align*}
            \sigma(z) = \frac{1}{1 + \exp (-z)} = \begin{cases} 
                1 &amp; &#x82E5;~z \rightarrow \infty \\ 
                0 &amp; &#x82E5;~z \rightarrow -\infty
            \end{cases}
        \end{align*}
    $$
</div>
<br>
<p>Logistic &#x51FD;&#x6570;&#x8FDE;&#x7EED;&#x53EF;&#x5BFC;&#xFF0C;&#x5728;<span class="blue">&#x96F6;&#x5904;&#x5BFC;&#x6570;&#x6700;&#x5927;</span></p>
<div>
    $$
        \begin{align*}
            \sigma&apos;(z) &amp; = - \frac{- \exp (-z)}{(1 + \exp (-z))^2} \\
            &amp; = \frac{1}{1 + \exp (-z)} \frac{\exp (-z)}{1 + \exp (-z)} \\
            &amp; = \sigma(z) (1 - \sigma(z)) \\
            &amp; \leq \left( \frac{\sigma(z) + 1 - \sigma(z)}{2} \right)^2 = \frac{1}{4} \quad &#x5728;z = 0&#x65F6;&#x53D6;&#x7B49;&#x53F7;
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="426" class="slide " data-line="426" data-h="2" data-v="3">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">Tanh&#x51FD;&#x6570;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5C06;<span class="mathjax-exps">$\Rbb$</span>&#x201C;&#x6324;&#x538B;&#x201D;&#x5230;<span class="mathjax-exps">$[-1,1]$</span>&#xFF0C;<span class="blue">&#x8F93;&#x51FA;&#x96F6;&#x4E2D;&#x5FC3;&#x5316;</span>&#xFF0C;Logistic &#x51FD;&#x6570;&#x7684;&#x653E;&#x5927;&#x5E73;&#x79FB;</p>
<div>
    $$
        \begin{align*}
            \tanh(z) &amp; = \frac{\exp(z) - \exp(-z)}{\exp(z) + \exp(-z)} \\
            &amp; = \frac{1 - \exp(-2z)}{1 + \exp(-2z)} \\
            &amp; = 2 \sigma(2z) - 1 = \begin{cases} 
                1 &amp; &#x82E5;~z \rightarrow \infty \\ 
                -1 &amp; &#x82E5;~z \rightarrow -\infty
            \end{cases}
        \end{align*}
    $$
</div>
<br>
<p>&#x6027;&#x8D28;&#xFF1A;</p>
<ul>
<li>&#x8FDE;&#x7EED;&#x53EF;&#x5BFC;<span class="mathjax-exps">$\tanh&apos;(z) = 2 (\sigma(2z))&apos; = 4 \sigma(2z) (1 - \sigma(2z))$</span></li>
<li>&#x8F93;&#x51FA;&#x96F6;&#x4E2D;&#x5FC3;&#x5316;&#x4F7F;&#x5F97;&#x540E;&#x4E00;&#x5C42;&#x7684;&#x8F93;&#x5165;<span class="mathjax-exps">$\wv^\top \av + \bv$</span>&#x4E5F;&#x5728;&#x96F6;&#x9644;&#x8FD1;&#xFF0C;&#x800C; Logistic &#x51FD;&#x6570;&#x5728;&#x96F6;&#x5904;&#x5BFC;&#x6570;&#x6700;&#x5927;&#xFF0C;&#x68AF;&#x5EA6;&#x4E0B;&#x964D;&#x66F4;&#x65B0;&#x6548;&#x7387;&#x8F83;&#x9AD8;&#xFF0C;Logistic &#x51FD;&#x6570;&#x8F93;&#x51FA;&#x6052;&#x4E3A;&#x6B63;&#xFF0C;&#x4F1A;&#x8FDB;&#x4E00;&#x6B65;&#x51CF;&#x6162;&#x68AF;&#x5EA6;&#x4E0B;&#x964D;&#x7684;&#x6536;&#x655B;&#x901F;&#x5EA6;</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="466" class="slide " data-line="466" data-h="3" data-v="0">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">ReLU</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>ReLU &#x5168;&#x79F0;&#x53EB;&#x4FEE;&#x6B63;&#x7EBF;&#x6027;&#x5355;&#x5143;(Rectified Linear Unit)&#xFF0C;&#x5B9A;&#x4E49;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \relu(z) &amp; = \begin{cases}
                z &amp; &#x82E5;~z \geq 0 \\ 0 &amp; &#x82E5;~z &lt; 0
            \end{cases} \\
            &amp; = \max \{ 0, z \} 
        \end{align*}
    $$
</div>
<br>
<p>&#x4F18;&#x70B9;</p>
<ul>
<li>&#x8BA1;&#x7B97;&#x53EA;&#x6D89;&#x53CA;&#x52A0;&#x6CD5;&#x3001;&#x4E58;&#x6CD5;&#x548C;&#x6BD4;&#x8F83;&#x64CD;&#x4F5C;&#xFF0C;&#x975E;&#x5E38;&#x9AD8;&#x6548;</li>
<li>&#x751F;&#x7269;&#x89E3;&#x91CA;&#xFF1A;&#x5355;&#x4FA7;&#x6291;&#x5236;&#xFF0C;&#x5BBD;&#x5174;&#x594B;&#x8FB9;&#x754C;&#xFF0C;&#x7A00;&#x758F;&#x5174;&#x594B;</li>
<li>&#x5728;<span class="mathjax-exps">$z &gt; 0$</span>&#x65F6;&#x5BFC;&#x6570;&#x6052;&#x4E3A;<span class="mathjax-exps">$1$</span>&#xFF0C;&#x7F13;&#x89E3;&#x4E86;<span class="blue">&#x68AF;&#x5EA6;&#x6D88;&#x5931;</span>&#x95EE;&#x9898;</li>
</ul>
<br>
<p>&#x7F3A;&#x70B9;</p>
<ul>
<li>&#x8F93;&#x51FA;&#x975E;&#x96F6;&#x4E2D;&#x5FC3;&#x5316;</li>
<li>&#x6B7B;&#x4EA1; ReLU &#x95EE;&#x9898;</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="512" class="slide " data-line="512" data-h="3" data-v="1">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6B7B;&#x4EA1;ReLU&#x95EE;&#x9898;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x7531;&#x94FE;&#x5F0F;&#x6CD5;&#x5219;&#x6709;</p>
<div>
    $$
        \begin{align*}
            \frac{\partial \relu(\wv^\top \xv + b)}{\partial \wv} &amp; = \frac{\partial \relu(\wv^\top \xv + b)}{\partial (\wv^\top \xv + b)} \frac{\partial (\wv^\top \xv + b)}{\partial \wv} \\
            &amp; = \frac{\partial \max \{ 0, \wv^\top \xv + b \}}{\partial (\wv^\top \xv + b)} \xv^\top \\
            &amp; = 1_{\wv^\top \xv + b \geq 0} \xv^\top
        \end{align*}
    $$
</div>
<p>&#x5982;&#x679C;&#x7B2C;&#x4E00;&#x9690;&#x85CF;&#x5C42;&#x4E2D;&#x7684;&#x67D0;&#x4E2A;&#x795E;&#x7ECF;&#x5143;&#x7684;&#x6743;&#x91CD;&#x5411;&#x91CF;<span class="mathjax-exps">$\wv$</span>&#x8BBE;&#x7F6E;&#x4E0D;&#x5F53;&#xFF0C;&#x4F7F;&#x5F97;&#x5BF9;&#x4EFB;&#x610F;<span class="mathjax-exps">$\xv$</span>&#x6709;<span class="mathjax-exps">$\wv^\top \xv + b &lt; 0$</span>&#xFF0C;&#x90A3;&#x4E48;&#x5176;&#x5173;&#x4E8E;<span class="mathjax-exps">$\wv$</span>&#x7684;&#x68AF;&#x5EA6;&#x5C06;&#x4E3A;&#x96F6;&#xFF0C;&#x5728;&#x4EE5;&#x540E;&#x7684;&#x8BAD;&#x7EC3;&#x8FC7;&#x7A0B;&#x4E2D;<span class="mathjax-exps">$\wv$</span>&#x6C38;&#x8FDC;&#x4E0D;&#x4F1A;&#x88AB;&#x66F4;&#x65B0;</p>
<br>
<p>&#x65B9;&#x6848;&#xFF1A;&#x5E26;&#x6CC4;&#x6F0F;&#x7684; ReLU&#xFF0C;&#x5E26;&#x53C2;&#x6570;&#x7684; ReLU&#xFF0C;ELU&#xFF0C;Softplus</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="548" class="slide " data-line="548" data-h="3" data-v="2">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">ReLU&#x53D8;&#x4F53;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5E26;&#x6CC4;&#x6F0F;&#x7684; ReLU&#xFF1A;&#x5F53;<span class="mathjax-exps">$\wv^\top \xv + b &lt; 0$</span>&#x65F6;&#x4E5F;&#x6709;&#x975E;&#x96F6;&#x68AF;&#x5EA6;</p>
<div>
    $$
        \begin{align*}
            \lrelu(z) &amp; = \begin{cases}
                z &amp; &#x82E5;~z \geq 0 \\ \gamma z &amp; &#x82E5;~z &lt; 0
            \end{cases} \\
            &amp; = \max \{ 0, z \} + \gamma \min \{ 0, z \} \overset{\gamma &lt; 1}{=} \max \{ z, \gamma z \}
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;&#x659C;&#x7387;<span class="mathjax-exps">$\gamma$</span>&#x662F;&#x4E00;&#x4E2A;&#x5F88;&#x5C0F;&#x7684;&#x5E38;&#x6570;&#xFF0C;&#x6BD4;&#x5982;<span class="mathjax-exps">$0.01$</span></p>
<br>
<p>&#x5E26;&#x53C2;&#x6570;&#x7684; ReLU&#xFF1A;&#x659C;&#x7387;<span class="mathjax-exps">$\gamma_i$</span>&#x53EF;&#x5B66;&#x4E60;</p>
<div>
    $$
        \begin{align*}
            \prelu(z) &amp; = \begin{cases}
                z &amp; &#x82E5;~z \geq 0 \\ \gamma_i z &amp; &#x82E5;~z &lt; 0
            \end{cases} \\
            &amp; = \max \{ 0, z \} + \gamma_i \min \{ 0, z \} 
        \end{align*}
    $$
</div>
<p>&#x53EF;&#x4EE5;&#x4E0D;&#x540C;&#x795E;&#x7ECF;&#x5143;&#x5177;&#x6709;&#x4E0D;&#x540C;&#x7684;&#x53C2;&#x6570;&#xFF0C;&#x4E5F;&#x53EF;&#x4EE5;&#x4E00;&#x7EC4;&#x795E;&#x7ECF;&#x5143;&#x5171;&#x4EAB;&#x4E00;&#x4E2A;&#x53C2;&#x6570;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="598" class="slide " data-line="598" data-h="3" data-v="3">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">ReLU&#x53D8;&#x4F53;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>ELU &#x5168;&#x79F0;&#x53EB;&#x6307;&#x6570;&#x7EBF;&#x6027;&#x5355;&#x5143;(Exponential Linear Unit)</p>
<div>
    $$
        \begin{align*}
            \elu(z) &amp; = \begin{cases}
                z &amp; &#x82E5;~z \geq 0 \\ \gamma (\exp(z) - 1) &amp; &#x82E5;~z &lt; 0
            \end{cases} \\
            &amp; = \max \{ 0, z \} + \min \{ 0, \gamma (\exp(z) - 1) \}
        \end{align*}
    $$
</div>
<br>
<p>Softplus &#x51FD;&#x6570;&#x53EF;&#x4EE5;&#x770B;&#x4F5C; ReLU &#x7684;&#x5E73;&#x6ED1;&#x7248;&#x672C;&#xFF1A;</p>
<div>
    $$
        \begin{align*}
            \softplus(z) = \log (1 + \exp(z))
        \end{align*}
    $$
</div>
<p>&#x5176;&#x5BFC;&#x6570;&#x4E3A; Logistic &#x51FD;&#x6570;</p>
<div>
    $$
        \begin{align*}
            \softplus&apos;(z) = \frac{\exp(z)}{1 + \exp(z)} = \frac{1}{1 + \exp(-z)}
        \end{align*}
    $$
</div>
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    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="651" class="slide " data-line="651" data-h="3" data-v="4">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">ReLU&#x65CF;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
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" style="margin-top:2rem" height="420px">
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section></section><section><section data-notes lineno="671" class="slide " data-line="671" data-h="4" data-v="0">
<div class="multi_column">
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        <h5 class="title">Swish&#x51FD;&#x6570;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>Swish &#x51FD;&#x6570;&#x662F;&#x4E00;&#x79CD;&#x81EA;&#x95E8;&#x63A7;&#x6FC0;&#x6D3B;&#x51FD;&#x6570;&#xFF1A;</p>
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<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\beta$</span>&#x662F;&#x53EF;&#x5B66;&#x4E60;&#x7684;&#x53C2;&#x6570;&#x6216;&#x4E00;&#x4E2A;&#x56FA;&#x5B9A;&#x8D85;&#x53C2;&#x6570;</p>
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<li>&#x5F53;<span class="mathjax-exps">$\sigma (\beta z)$</span>&#x63A5;&#x8FD1;&#x4E8E;<span class="mathjax-exps">$1$</span>&#x65F6;&#xFF0C;&#x95E8;&#x5904;&#x4E8E;&#x201C;&#x5F00;&#x201D;&#x72B6;&#x6001;&#xFF0C;&#x6FC0;&#x6D3B;&#x51FD;&#x6570;&#x7684;&#x8F93;&#x51FA;&#x8FD1;&#x4F3C;&#x4E8E;<span class="mathjax-exps">$z$</span>&#x672C;&#x8EAB;</li>
<li>&#x5F53;<span class="mathjax-exps">$\sigma (\beta z)$</span>&#x63A5;&#x8FD1;&#x4E8E;<span class="mathjax-exps">$0$</span>&#x65F6;&#xFF0C;&#x95E8;&#x5904;&#x4E8E;&#x201C;&#x5173;&#x201D;&#x72B6;&#x6001;&#xFF0C;&#x6FC0;&#x6D3B;&#x51FD;&#x6570;&#x7684;&#x8F93;&#x51FA;&#x8FD1;&#x4F3C;&#x4E8E;<span class="mathjax-exps">$0$</span></li>
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        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</div>
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        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
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<li>&#x82E5;<span class="mathjax-exps">$y \in [C]$</span>&#xFF0C;&#x6700;&#x540E;&#x4E00;&#x5C42;&#x9700;<span class="mathjax-exps">$C$</span>&#x4E2A;&#x795E;&#x7ECF;&#x5143;&#xFF0C;&#x5E76;&#x91C7;&#x7528; Softmax &#x6FC0;&#x6D3B;&#x51FD;&#x6570;</li>
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
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</div>
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    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
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<p>&#x4F20;&#x7EDF;&#x673A;&#x5668;&#x5B66;&#x4E60;&#xFF1A;&#x7279;&#x5F81;&#x5904;&#x7406;&#x548C;&#x5B66;&#x4E60;&#x4E24;&#x9636;&#x6BB5;&#x5206;&#x5F00;&#x8FDB;&#x884C;</p>
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<!-- 预测 -->
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</p><div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="934" class="slide " data-line="934" data-h="5" data-v="2">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6C42;&#x89E3;&#x53C2;&#x6570;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x4E0D;&#x59A8;&#x8BBE;&#x91C7;&#x7528;&#x4EA4;&#x53C9;&#x71B5;&#x635F;&#x5931;&#x51FD;&#x6570;&#xFF0C;&#x5BF9;&#x4E8E;&#x6837;&#x672C;<span class="mathjax-exps">$(\xv, y)$</span>&#xFF0C;&#x5176;&#x635F;&#x5931;&#x51FD;&#x6570;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \Lcal (\yv, \hat{\yv}) = - \yv \log \hat{\yv}
        \end{align*}
    $$
</div>
<br>
<p>&#x4F18;&#x5316;&#x76EE;&#x6807;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\Wv, \bv} ~ \frac{\lambda}{m} \sum_{i \in [m]} \Lcal (\yv_i, \hat{\yv}_i) + \frac{1}{2} \| \Wv \|_F^2
        \end{align*}
    $$
</div>
<p>&#x68AF;&#x5EA6;&#x4E0B;&#x964D;</p>
<div>
    $$
        \begin{align*}
            \Wv &amp; \leftarrow \Wv - \eta \left( \frac{\lambda}{m} \sum_{i \in [m]} \class{yellow}{\frac{\partial \Lcal (\yv_i, \hat{\yv}_i)}{\partial \Wv}} + \Wv \right) \\
            \bv &amp; \leftarrow \bv - \eta \cdot \frac{\lambda}{m} \sum_{i \in [m]} \class{yellow}{\frac{\partial \Lcal (\yv_i, \hat{\yv}_i)}{\partial \bv}}
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="985" class="slide " data-line="985" data-h="6" data-v="0">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x77E9;&#x9635;&#x6C42;&#x5BFC;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5148;&#x8BC1;&#x660E;&#x51E0;&#x4E2A;&#x5BF9;&#x77E9;&#x9635;&#x6C42;&#x5BFC;&#x7684;&#x7ED3;&#x8BBA;</p>
<div>
    $$
        \begin{align*}
            \underbrace{\zv}_{n \times 1} = \underbrace{\Wv}_{n \times m} ~ \underbrace{\av}_{m \times 1} + \underbrace{\bv}_{n \times 1}
        \end{align*}
    $$
</div>
<p>&#x8BB0;<span class="mathjax-exps">$\Wv = [\wv_1, \ldots, \wv_n]^\top = [\wv_1^\top; \ldots; \wv_n^\top]$</span>&#xFF0C;&#x5199;&#x6210;&#x65B9;&#x7A0B;&#x7EC4;&#x7684;&#x5F62;&#x5F0F;</p>
<div>
    $$
        \begin{align*}
            z_1 &amp; = \wv_1^\top \av + b_1 = w_{11} a_1 + w_{12} a_2 + \cdots + w_{1m} a_m + b_1 \\
            z_2 &amp; = \wv_2^\top \av + b_2 = w_{21} a_1 + w_{22} a_2 + \cdots + w_{2m} a_m + b_2 \\
            &amp; \quad \vdots \\
            z_n &amp; = \wv_n^\top \av + b_n = w_{n1} a_1 + w_{n2} a_2 + \cdots + w_{nm} a_m + b_n \\
        \end{align*}
    $$
</div>
<p>&#x4E8E;&#x662F;</p>
<div>
    $$
        \begin{align*}
            \frac{\partial z_j}{\partial b_i} = 1_{j=i} \Longrightarrow \frac{\partial \zv}{\partial b_i} = \ev_i \Longrightarrow \frac{\partial \zv}{\partial \bv} = [\ev_1, \ev_2, \ldots, \ev_n] = \Iv 
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1036" class="slide " data-line="1036" data-h="6" data-v="1">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x77E9;&#x9635;&#x6C42;&#x5BFC;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5148;&#x8BC1;&#x660E;&#x51E0;&#x4E2A;&#x5BF9;&#x77E9;&#x9635;&#x6C42;&#x5BFC;&#x7684;&#x7ED3;&#x8BBA;</p>
<div>
    $$
        \begin{align*}
            \underbrace{\zv}_{n \times 1} = \underbrace{\Wv}_{n \times m} ~ \underbrace{\av}_{m \times 1} + \underbrace{\bv}_{n \times 1}
        \end{align*}
    $$
</div>
<p>&#x8BB0;<span class="mathjax-exps">$\Wv = [\wv_1, \ldots, \wv_n]^\top = [\wv_1^\top; \ldots; \wv_n^\top]$</span>&#xFF0C;&#x5199;&#x6210;&#x65B9;&#x7A0B;&#x7EC4;&#x7684;&#x5F62;&#x5F0F;</p>
<div>
    $$
        \begin{align*}
            z_1 &amp; = \wv_1^\top \av + b_1 = w_{11} a_1 + w_{12} a_2 + \cdots + w_{1m} a_m + b_1 \\
            z_2 &amp; = \wv_2^\top \av + b_2 = w_{21} a_1 + w_{22} a_2 + \cdots + w_{2m} a_m + b_2 \\
            &amp; \quad \vdots \\
            z_n &amp; = \wv_n^\top \av + b_n = w_{n1} a_1 + w_{n2} a_2 + \cdots + w_{nm} a_m + b_n \\
        \end{align*}
    $$
</div>
<p>&#x4E8E;&#x662F;</p>
<div>
    $$
        \begin{align*}
            \frac{\partial z_j}{\partial a_i} = w_{ji} \Longrightarrow \frac{\partial z_j}{\partial \av} = [w_{j1}, \ldots, w_{jm}] = \wv_j^\top \Longrightarrow \frac{\partial \zv}{\partial \av} = \Wv
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1087" class="slide " data-line="1087" data-h="6" data-v="2">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x77E9;&#x9635;&#x6C42;&#x5BFC;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5148;&#x8BC1;&#x660E;&#x51E0;&#x4E2A;&#x5BF9;&#x77E9;&#x9635;&#x6C42;&#x5BFC;&#x7684;&#x7ED3;&#x8BBA;</p>
<div>
    $$
        \begin{align*}
            \underbrace{\zv}_{n \times 1} = \underbrace{\Wv}_{n \times m} ~ \underbrace{\av}_{m \times 1} + \underbrace{\bv}_{n \times 1}
        \end{align*}
    $$
</div>
<p>&#x8BB0;<span class="mathjax-exps">$\Wv = [\wv_1, \ldots, \wv_n]^\top = [\wv_1^\top; \ldots; \wv_n^\top]$</span>&#xFF0C;&#x5199;&#x6210;&#x65B9;&#x7A0B;&#x7EC4;&#x7684;&#x5F62;&#x5F0F;</p>
<div>
    $$
        \begin{align*}
            z_1 &amp; = \wv_1^\top \av + b_1 = w_{11} a_1 + w_{12} a_2 + \cdots + w_{1m} a_m + b_1 \\
            z_2 &amp; = \wv_2^\top \av + b_2 = w_{21} a_1 + w_{22} a_2 + \cdots + w_{2m} a_m + b_2 \\
            &amp; \quad \vdots \\
            z_n &amp; = \wv_n^\top \av + b_n = w_{n1} a_1 + w_{n2} a_2 + \cdots + w_{nm} a_m + b_n \\
        \end{align*}
    $$
</div>
<p>&#x4E8E;&#x662F;</p>
<div>
    $$
        \begin{align*} 
            \frac{\partial z_j}{\partial w_{ik}} = 1_{j=i} a_k &amp; \Longrightarrow \frac{\partial z_j}{\partial \wv_i} = 1_{j=i} \av^\top \Longrightarrow \frac{\partial z_j}{\partial \Wv} = [\underbrace{\zerov, \ldots, \av, \ldots, \zerov}_{~&#x7B2C;j&#x5217;&#x4E3A;\av}]
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1138" class="slide " data-line="1138" data-h="6" data-v="3">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6C42;&#x89E3;&#x68AF;&#x5EA6;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5BF9;&#x4E8E;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#xFF1A;<span class="mathjax-exps">$\underbrace{\zv_l}_{n_l \times 1} = \underbrace{\Wv_l}_{n_l \times n_{l-1}} ~ \underbrace{\av_{l-1}}_{n_{l-1} \times 1} + \underbrace{\bv_l}_{n_l \times 1}$</span></p>
<p>&#x8BB0;<span class="mathjax-exps">$\Wv_l = [\wv_1, \ldots, \wv_n]^\top = [\wv_1^\top; \ldots; \wv_n^\top]$</span>&#xFF0C;&#x7EFC;&#x4E0A;&#x6709;</p>
<div>
    $$
        \begin{align*}
            \frac{\partial \zv_l}{\partial \bv_l} = \Iv, \quad \frac{\partial \zv_l}{\partial \av_{l-1}} = \Wv_l, \quad \frac{\partial [\zv_l]_j}{\partial \Wv_l} = [\zerov, \ldots, \av_{l-1}, \ldots, \zerov]
        \end{align*}
    $$
</div>
<p>&#x8BB0;<span class="mathjax-exps">$\deltav_l^\top = \partial \Lcal (\yv, \hat{\yv}) / \partial \zv_l \in \Rbb^{n_l}$</span>&#x4E3A;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#x7684;&#x8BEF;&#x5DEE;&#x9879;&#xFF0C;&#x7531;&#x94FE;&#x5F0F;&#x6CD5;&#x5219;</p>
<div>
    $$
        \begin{align*}
            \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \bv_l} &amp; = \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \zv_l} \frac{\partial \zv_l}{\partial \bv_l} = \deltav_l^\top \Iv = \deltav_l^\top \\
            \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \av_{l-1}} &amp; = \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \zv_l} \frac{\partial \zv_l}{\partial \av_{l-1}} = \deltav_l^\top \Wv_l
        \end{align*}
    $$
</div>
<p>&#x8FD9;&#x91CC;&#x5BF9;<span class="mathjax-exps">$\av_{l-1}$</span>&#x4E5F;&#x8981;&#x6C42;&#x5BFC;&#x662F;&#x56E0;&#x4E3A;<span class="mathjax-exps">$\av_{l-1}$</span>&#x4E0E;&#x524D;&#x4E00;&#x5C42;&#x7684;&#x53D8;&#x91CF;<span class="mathjax-exps">$\Wv_{l-1}$</span>&#x662F;&#x76F8;&#x5173;&#x7684;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="1181" class="slide " data-line="1181" data-h="7" data-v="0">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6C42;&#x89E3;&#x68AF;&#x5EA6;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5BF9;&#x4E8E;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#xFF1A;<span class="mathjax-exps">$\underbrace{\zv_l}_{n_l \times 1} = \underbrace{\Wv_l}_{n_l \times n_{l-1}} ~ \underbrace{\av_{l-1}}_{n_{l-1} \times 1} + \underbrace{\bv_l}_{n_l \times 1}$</span></p>
<p>&#x8BB0;<span class="mathjax-exps">$\Wv_l = [\wv_1, \ldots, \wv_n]^\top = [\wv_1^\top; \ldots; \wv_n^\top]$</span>&#xFF0C;&#x7EFC;&#x4E0A;&#x6709;</p>
<div>
    $$
        \begin{align*}
            \frac{\partial \zv_l}{\partial \bv_l} = \Iv, \quad \frac{\partial \zv_l}{\partial \av_{l-1}} = \Wv_l, \quad \frac{\partial [\zv_l]_j}{\partial \Wv_l} = [\zerov, \ldots, \av_{l-1}, \ldots, \zerov]
        \end{align*}
    $$
</div>
<p>&#x8BB0;<span class="mathjax-exps">$\deltav_l^\top = \partial \Lcal (\yv, \hat{\yv}) / \partial \zv_l \in \Rbb^{n_l}$</span>&#x4E3A;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#x7684;&#x8BEF;&#x5DEE;&#x9879;&#xFF0C;&#x7531;&#x94FE;&#x5F0F;&#x6CD5;&#x5219;</p>
<div>
    $$
        \begin{align*}
            \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \Wv_l} &amp; = \sum_{j \in [n_l]} \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial [\zv_l]_j} \frac{\partial [\zv_l]_j}{\partial \Wv_l} = \sum_{j \in [n_l]} [\deltav_l]_j [\zerov, \ldots, \av_{l-1}, \ldots, \zerov] \\
            &amp; = [\av_{l-1} [\deltav_l]_1, \zerov, \ldots, \zerov] + \cdots + [\zerov, \ldots, \zerov, \av_{l-1} [\deltav_l]_{n_l}] \\
            &amp; = [\av_{l-1} [\deltav_l]_1, \ldots, \av_{l-1} [\deltav_l]_{n_l}] \\
            &amp; = \av_{l-1} \deltav_l^\top
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1224" class="slide " data-line="1224" data-h="7" data-v="1">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6C42;&#x89E3;&#x68AF;&#x5EA6;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#xFF1A;</p>
<div>
    $$
        \begin{align*}
            \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \Wv_l} = \av_{l-1} \deltav_l^\top, \quad \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \bv_l} = \deltav_l^\top, \quad \frac{\partial \zv_l}{\partial \av_{l-1}} = \Wv_l
        \end{align*}
    $$
</div>
<p>&#x7B2C;<span class="mathjax-exps">$l+1$</span>&#x5C42;&#xFF1A;<span class="mathjax-exps">$\zv_{l+1} = \Wv_{l+1} ~ \av_l + \bv_{l+1}$</span>&#xFF0C;&#x5176;&#x4E2D;<span class="mathjax-exps">$\av_l = h_l (\zv_l)$</span>&#xFF0C;&#x7531;&#x94FE;&#x5F0F;&#x6CD5;&#x5219;</p>
<div>
    $$
        \begin{align*}
            \deltav_l^\top = \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \zv_l} = \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \zv_{l+1}} \frac{\partial \zv_{l+1}}{\partial \av_l} \frac{\partial \av_l}{\partial \zv_l} = \deltav_{l+1}^\top \Wv_{l+1} \diag (h_l&apos;(\zv_l))
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$h_l&apos;(\zv_l)$</span>&#x662F;&#x6FC0;&#x6D3B;&#x51FD;&#x6570;&#x7684;&#x5BFC;&#x6570;</p>
<br>
<p>&#x8BEF;&#x5DEE;<span class="blue">&#x53CD;&#x5411;&#x4F20;&#x64AD;</span>&#xFF1A;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#x7684;&#x8BEF;&#x5DEE;&#x9879;&#x53EF;&#x4EE5;&#x901A;&#x8FC7;&#x7B2C;<span class="mathjax-exps">$l+1$</span>&#x5C42;&#x7684;&#x8BEF;&#x5DEE;&#x9879;&#x8BA1;&#x7B97;&#x5F97;&#x5230;</p>
<p>&#x5269;&#x4E0B;&#x53EA;&#x9700;&#x6C42;&#x6700;&#x540E;&#x4E00;&#x5C42;&#x7684;&#x8BEF;&#x5DEE;&#x9879;</p>
<div>
    $$
        \begin{align*}
            \deltav_L^\top = \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \zv_L} = \frac{\partial (-\yv \log \hat{\yv})}{\partial \zv_L} = - \frac{\partial \yv \log h_L(\zv_L)}{\partial \zv_L}
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1278" class="slide " data-line="1278" data-h="7" data-v="2">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x53CD;&#x5411;&#x4F20;&#x64AD;&#x7B97;&#x6CD5;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x8F93;&#x5165;&#xFF1A;&#x8BAD;&#x7EC3;&#x96C6;<span class="mathjax-exps">$\Scal$</span>&#xFF0C;&#x9A8C;&#x8BC1;&#x96C6;<span class="mathjax-exps">$\Vcal$</span>&#xFF0C;&#x4EE5;&#x53CA;&#x76F8;&#x5173;&#x8D85;&#x53C2;&#x6570;</p>
<ol>
<li>&#x968F;&#x673A;&#x521D;&#x59CB;&#x5316;<span class="mathjax-exps">$\Wv$</span>&#x3001;<span class="mathjax-exps">$\bv$</span></li>
<li>repeat</li>
<li>&#x2003;&#x2003;&#x5BF9;&#x8BAD;&#x7EC3;&#x96C6;<span class="mathjax-exps">$\Scal$</span>&#x4E2D;&#x7684;&#x6837;&#x672C;&#x968F;&#x673A;&#x91CD;&#x6392;&#x5E8F;</li>
<li>&#x2003;&#x2003;for <span class="mathjax-exps">$i = 1, \ldots, m$</span> do</li>
<li>&#x2003;&#x2003;&#x2003;&#x2003;&#x83B7;&#x53D6;&#x6837;&#x672C;<span class="mathjax-exps">$(\xv_i, y_i)$</span></li>
<li>&#x2003;&#x2003;&#x2003;&#x2003;&#x524D;&#x5411;&#x8BA1;&#x7B97;&#x6BCF;&#x4E00;&#x5C42;&#x7684;<span class="mathjax-exps">$\zv_l$</span>&#x3001;<span class="mathjax-exps">$\av_l$</span>&#xFF0C;&#x76F4;&#x5230;&#x6700;&#x540E;&#x4E00;&#x5C42;</li>
<li>&#x2003;&#x2003;&#x2003;&#x2003;&#x53CD;&#x5411;&#x4F20;&#x64AD;&#x8BA1;&#x7B97;&#x6BCF;&#x4E00;&#x5C42;&#x7684;&#x8BEF;&#x5DEE;&#x9879;<span class="mathjax-exps">$\deltav_l$</span></li>
<li>&#x2003;&#x2003;&#x2003;&#x2003;&#x91C7;&#x7528;&#x68AF;&#x5EA6;&#x4E0B;&#x964D;&#x66F4;&#x65B0;<span class="mathjax-exps">$\Wv$</span>&#x3001;<span class="mathjax-exps">$\bv$</span></li>
<li>&#x2003;&#x2003;end</li>
<li>until &#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x6A21;&#x578B;&#x5728;&#x9A8C;&#x8BC1;&#x96C6;<span class="mathjax-exps">$\Vcal$</span>&#x4E0A;&#x7684;&#x9519;&#x8BEF;&#x7387;&#x4E0D;&#x518D;&#x4E0B;&#x964D;</li>
</ol>
<p>&#x8F93;&#x51FA;&#xFF1A;<span class="mathjax-exps">$\Wv$</span>&#x3001;<span class="mathjax-exps">$\bv$</span></p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="1311" class="slide " data-line="1311" data-h="8" data-v="0">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x68AF;&#x5EA6;&#x6D88;&#x5931;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x4E2D;&#x8BEF;&#x5DEE;&#x53CD;&#x5411;&#x4F20;&#x64AD;&#x7684;&#x8FED;&#x4EE3;&#x516C;&#x5F0F;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \deltav_l^\top = \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \zv_l} = \frac{\partial \Lcal (\yv, \hat{\yv})}{\partial \zv_{l+1}} \frac{\partial \zv_{l+1}}{\partial \av_l} \frac{\partial \av_l}{\partial \zv_l} = \deltav_{l+1}^\top \Wv_{l+1} \diag (h_l&apos;(\zv_l))
        \end{align*}
    $$
</div>
<br>
<p>&#x5BF9;&#x4E8E; Sigmoid &#x578B;&#x6FC0;&#x6D3B;&#x51FD;&#x6570;</p>
<ul>
<li><span class="mathjax-exps">$\sigma&apos;(z) = \sigma(z) (1 - \sigma(z)) \leq \frac{1}{4}$</span></li>
<li><span class="mathjax-exps">$\tanh&apos;(z) = 4 \sigma(2z) (1 - \sigma(2z)) \leq 4 \cdot \frac{1}{4} = 1$</span></li>
</ul>
<p>&#x8BEF;&#x5DEE;&#x6BCF;&#x4F20;&#x64AD;&#x4E00;&#x5C42;&#x90FD;&#x4F1A;&#x4E58;&#x4EE5;&#x4E00;&#x4E2A;&#x5C0F;&#x4E8E;&#x7B49;&#x4E8E;<span class="mathjax-exps">$1$</span>&#x7684;&#x7CFB;&#x6570;&#xFF0C;&#x5F53;&#x7F51;&#x7EDC;&#x5C42;&#x6570;&#x5F88;&#x6DF1;&#x65F6;&#xFF0C;&#x68AF;&#x5EA6;&#x4F1A;&#x4E0D;&#x65AD;&#x8870;&#x51CF;&#x751A;&#x81F3;&#x6D88;&#x5931;&#xFF0C;&#x4F7F;&#x5F97;&#x6574;&#x4E2A;&#x7F51;&#x7EDC;&#x5F88;&#x96BE;&#x8BAD;&#x7EC3;</p>
<br>
<p>&#x65B9;&#x6848;&#xFF1A;&#x4F7F;&#x7528;&#x5BFC;&#x6570;&#x6BD4;&#x8F83;&#x5927;&#x7684;&#x6FC0;&#x6D3B;&#x51FD;&#x6570;&#xFF0C;&#x6BD4;&#x5982; ReLU</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="truedata-notes" =true lineno="1352" class="slide " data-line="1352" data-h="8" data-v="1">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6B8B;&#x5DEE;&#x7F51;&#x7EDC;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
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<p>&#x6B8B;&#x5DEE;&#x6A21;&#x5757;&#xFF1A;<span class="mathjax-exps">$\zv_l = \av_{l-1} + \class{yellow}{\Uv_2 \cdot h_1(\Uv_1 \cdot \av_{l-1} + \cv_1) + \cv_2)} = \av_{l-1} + \class{yellow}{f(\av_{l-1})}$</span></p>
<p>&#x5047;&#x8BBE;<span class="mathjax-exps">$\av_l = \zv_l$</span>&#xFF0C;&#x5373;&#x6B8B;&#x5DEE;&#x6A21;&#x5757;&#x8F93;&#x51FA;&#x4E0D;&#x4F7F;&#x7528;&#x6FC0;&#x6D3B;&#x51FD;&#x6570;&#xFF0C;&#x5BF9;<span class="mathjax-exps">$\forall t \in [l]$</span>&#x6709;</p>
<div>
    $$
        \begin{align*}
            \av_l &amp; = \av_{l-1} + f(\av_{l-1}) = \av_{l-2} + f(\av_{l-2}) + f(\av_{l-1}) \\ 
            &amp; = \cdots = \av_{l-t} + \sum_{i=l-t}^{l-1} f(\av_i)
        \end{align*}
    $$
</div>
<p>&#x4F4E;&#x5C42;&#x8F93;&#x5165;&#x53EF;&#x4EE5;<span class="blue">&#x6052;&#x7B49;</span>&#x4F20;&#x64AD;&#x5230;&#x4EFB;&#x610F;&#x9AD8;&#x5C42;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="truedata-notes" =true lineno="1387" class="slide " data-line="1387" data-h="8" data-v="2">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6B8B;&#x5DEE;&#x7F51;&#x7EDC;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<div>
    $$
        \begin{align*}
            \av_l = \av_{l-t} + \sum_{i=l-t}^{l-1} f(\av_i)
        \end{align*}
    $$
</div>
<p>&#x7531;&#x94FE;&#x5F0F;&#x6CD5;&#x5219;&#x6709;</p>
<div>
    $$
        \begin{align*}
            \frac{\partial \Lcal}{\partial \av_{l-t}} &amp; = \frac{\partial \Lcal}{\partial \av_l} \frac{\partial \av_l}{\partial \av_{l-t}} = \frac{\partial \Lcal}{\partial \av_l} \left( \frac{\partial \av_{l-t}}{\partial \av_{l-t}} + \frac{\partial }{\partial \av_{l-t}} \sum_{i=l-t}^{l-1} f(\av_i) \right) \\
            &amp; = \frac{\partial \Lcal}{\partial \av_l} \left( \Iv + \frac{\partial }{\partial \av_{l-t}} \sum_{i=l-t}^{l-1} f(\av_i) \right) \\
            &amp; = \frac{\partial \Lcal}{\partial \av_l} + \frac{\partial \Lcal}{\partial \av_l} \left( \frac{\partial }{\partial \av_{l-t}} \sum_{i=l-t}^{l-1} f(\av_i) \right) 
        \end{align*}
    $$
</div>
<p>&#x9AD8;&#x5C42;&#x8BEF;&#x5DEE;&#x53EF;&#x4EE5;<span class="blue">&#x6052;&#x7B49;</span>&#x4F20;&#x64AD;&#x5230;&#x4EFB;&#x610F;&#x4F4E;&#x5C42;&#xFF0C;&#x68AF;&#x5EA6;&#x6D88;&#x5931;&#x5F97;&#x4EE5;&#x7F13;&#x89E3;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="1427" class="slide " data-line="1427" data-h="9" data-v="0">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x4F18;&#x5316;&#x7B97;&#x6CD5;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x901A;&#x5E38;&#x91C7;&#x7528;&#x5C0F;&#x6279;&#x91CF;&#x68AF;&#x5EA6;&#x4E0B;&#x964D;<span class="mathjax-exps">$\Wv_{t+1} \leftarrow \Wv_t - \eta_t \gv_t$</span>&#xFF0C;&#x5176;&#x4E2D;</p>
<div>
$$
    \begin{align*}
        \gv_t = \frac{1}{|\Bcal_t|} \sum_{i \in \Bcal_t} \frac{\partial \Lcal (\yv_i, \hat{\yv}_i)}{\partial \Wv}
    \end{align*}
$$
</div>
<br>
<p>&#x6279;&#x91CF;&#x5927;&#x5C0F;<span class="mathjax-exps">$|\Bcal_t|$</span>&#x4E0D;&#x5F71;&#x54CD;&#x968F;&#x673A;&#x68AF;&#x5EA6;&#x7684;&#x671F;&#x671B;&#xFF0C;&#x4F46;&#x4F1A;&#x5F71;&#x54CD;&#x65B9;&#x5DEE;</p>
<ul>
<li><span class="mathjax-exps">$|\Bcal_t|$</span>&#x8D8A;&#x5927;&#xFF0C;&#x65B9;&#x5DEE;&#x8D8A;&#x5C0F;&#xFF0C;&#x8BAD;&#x7EC3;&#x8D8A;&#x7A33;&#x5B9A;&#xFF0C;&#x53EF;&#x4EE5;&#x91C7;&#x7528;&#x8F83;&#x5927;&#x7684;&#x6B65;&#x957F;&#x52A0;&#x5FEB;&#x6536;&#x655B;</li>
<li><span class="mathjax-exps">$|\Bcal_t|$</span>&#x8D8A;&#x5C0F;&#xFF0C;&#x65B9;&#x5DEE;&#x8D8A;&#x5927;&#xFF0C;&#x9700;&#x91C7;&#x7528;&#x8F83;&#x5C0F;&#x7684;&#x6B65;&#x957F;&#xFF0C;&#x5426;&#x5219;&#x53EF;&#x80FD;&#x4E0D;&#x6536;&#x655B;</li>
</ul>
<br>
<p>&#x7EBF;&#x6027;&#x7F29;&#x653E;&#x89C4;&#x5219;&#xFF1A;<span class="mathjax-exps">$|\Bcal_t|$</span>&#x589E;&#x52A0;<span class="mathjax-exps">$k$</span>&#x500D;&#xFF0C;&#x6B65;&#x957F;&#x4E5F;&#x589E;&#x52A0;<span class="mathjax-exps">$k$</span>&#x500D;&#xFF0C;&#x4F46;&#x5F53;<span class="mathjax-exps">$|\Bcal_t|$</span>&#x7279;&#x522B;&#x5927;&#x65F6;&#xFF0C;&#x7EBF;&#x6027;&#x7F29;&#x653E;&#x4E5F;&#x8FD8;&#x662F;&#x4F1A;&#x51FA;&#x73B0;&#x8BAD;&#x7EC3;&#x4E0D;&#x7A33;&#x5B9A;</p>
<ul>
<li>&#x6B65;&#x957F;&#x8C03;&#x6574;</li>
<li>&#x66F4;&#x65B0;&#x65B9;&#x5411;&#x8C03;&#x6574;</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1469" class="slide " data-line="1469" data-h="9" data-v="1">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6B65;&#x957F;&#x8C03;&#x6574;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
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<!-- 周期性步长 -->
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<!-- 自适应步长 -->
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<!-- 分段常数衰减 -->
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<!-- 逆时衰减 -->
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<!-- 步长衰减&#45;&gt;逆时衰减 -->
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<!-- (自然)指数衰减 -->
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<!-- 步长衰减&#45;&gt;(自然)指数衰减 -->
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<!-- 步长衰减&#45;&gt;余弦衰减 -->
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<!-- 循环步长 -->
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<!-- 周期性步长&#45;&gt;循环步长 -->
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<!-- 带热重启的SGD -->
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<!-- 周期性步长&#45;&gt;带热重启的SGD -->
<g id="edge10" class="edge">
<title>&#x5468;&#x671F;&#x6027;&#x6B65;&#x957F;-&gt;&#x5E26;&#x70ED;&#x91CD;&#x542F;&#x7684;SGD</title>
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<!-- AdaGrad -->
<g id="node12" class="node">
<title>AdaGrad</title>
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<!-- 自适应步长&#45;&gt;AdaGrad -->
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<!-- RMSprop -->
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<title>RMSprop</title>
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<!-- 自适应步长&#45;&gt;RMSprop -->
<g id="edge12" class="edge">
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<!-- AdaDelta -->
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<title>AdaDelta</title>
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<!-- 自适应步长&#45;&gt;AdaDelta -->
<g id="edge13" class="edge">
<title>&#x81EA;&#x9002;&#x5E94;&#x6B65;&#x957F;-&gt;AdaDelta</title>
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</svg>
</p><div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1505" class="slide " data-line="1505" data-h="9" data-v="2">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6B65;&#x957F;&#x8870;&#x51CF;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x57FA;&#x672C;&#x60F3;&#x6CD5;&#xFF1A;&#x5728;&#x4E00;&#x5F00;&#x59CB;&#x8981;&#x4F7F;&#x7528;&#x5927;&#x6B65;&#x957F;&#x6765;&#x4FDD;&#x8BC1;&#x6536;&#x655B;&#x901F;&#x5EA6;&#xFF0C;&#x5728;&#x63A5;&#x8FD1;&#x6700;&#x4F18;&#x89E3;&#x65F6;&#x8981;&#x7528;&#x5C0F;&#x6B65;&#x957F;&#x907F;&#x514D;&#x6765;&#x56DE;&#x9707;&#x8361;&#xFF0C;&#x4E5F;&#x79F0;&#x4E3A;&#x6B65;&#x957F;&#x9000;&#x706B;</p>
<p>&#x8BBE;&#x521D;&#x59CB;&#x6B65;&#x957F;&#x4E3A;<span class="mathjax-exps">$\eta_0$</span>&#xFF0C;&#x7B2C;<span class="mathjax-exps">$t$</span>&#x6B21;&#x8FED;&#x4EE3;&#x65F6;&#x7684;&#x6B65;&#x957F;&#x4E3A;<span class="mathjax-exps">$\eta_t$</span>&#xFF0C;&#x5E38;&#x89C1;&#x7684;&#x8870;&#x51CF;&#x65B9;&#x5F0F;&#x4E3A;&#x6839;&#x636E;&#x8FED;&#x4EE3;&#x6B21;&#x6570;&#x8FDB;&#x884C;&#x8870;&#x51CF;</p>
<ul>
<li>&#x5206;&#x6BB5;&#x5E38;&#x6570;&#x8870;&#x51CF;&#xFF1A;&#x6BCF;&#x7ECF;&#x8FC7;<span class="mathjax-exps">$T_1, T_2, \ldots, T_n$</span>&#x6B21;&#x8FED;&#x4EE3;&#x6B65;&#x957F;&#x8870;&#x51CF;&#x4E3A;&#x539F;&#x6765;&#x7684;<span class="mathjax-exps">$\beta_1, \beta_2, \ldots, \beta_n$</span>&#x500D;&#xFF0C;&#x5176;&#x4E2D;<span class="mathjax-exps">$T_n$</span>&#x3001;<span class="mathjax-exps">$\beta_n &lt; 1$</span>&#x4E3A;&#x8D85;&#x53C2;&#x6570;</li>
<li>&#x9006;&#x65F6;&#x8870;&#x51CF;&#xFF1A;<span class="mathjax-exps">$\eta_t = \eta_0 / (1 + \beta * t)$</span>&#xFF0C;&#x5176;&#x4E2D;<span class="mathjax-exps">$\beta$</span>&#x4E3A;&#x8870;&#x51CF;&#x7387;</li>
<li>&#x6307;&#x6570;&#x8870;&#x51CF;&#xFF1A;<span class="mathjax-exps">$\eta_t = \eta_0 \beta^t$</span>&#xFF0C;&#x5176;&#x4E2D;<span class="mathjax-exps">$\beta &lt; 1$</span>&#x4E3A;&#x8870;&#x51CF;&#x7387;</li>
<li>&#x81EA;&#x7136;&#x6307;&#x6570;&#x8870;&#x51CF;&#xFF1A;<span class="mathjax-exps">$\eta_t = \eta_0 \exp(-\beta * t)$</span></li>
<li>&#x4F59;&#x5F26;&#x8870;&#x51CF;&#xFF1A;<span class="mathjax-exps">$\eta_t = \eta_0 (1 + \cos (t \pi /T) )$</span>&#xFF0C;&#x5176;&#x4E2D;<span class="mathjax-exps">$T$</span>&#x4E3A;&#x603B;&#x7684;&#x8FED;&#x4EE3;&#x6B21;&#x6570;</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1533" class="slide " data-line="1533" data-h="9" data-v="3">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6B65;&#x957F;&#x9884;&#x70ED;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x6B65;&#x957F;&#x9884;&#x70ED;&#xFF1A;</p>
<ul>
<li>&#x5728;&#x6700;&#x521D;&#x51E0;&#x8F6E;&#x8FED;&#x4EE3;&#x65F6;&#xFF0C;&#x91C7;&#x7528;&#x8F83;&#x5C0F;&#x7684;&#x6B65;&#x957F;</li>
<li>&#x7B49;&#x68AF;&#x5EA6;&#x4E0B;&#x964D;&#x5230;&#x4E00;&#x5B9A;&#x7A0B;&#x5EA6;&#x540E;&#x518D;&#x6062;&#x590D;&#x4E3A;&#x521D;&#x59CB;&#x6B65;&#x957F;</li>
</ul>
<br>
<p>&#x52A8;&#x673A;&#xFF1A;&#x5728;&#x8BAD;&#x7EC3;&#x521D;&#x59CB;&#x9636;&#x6BB5;&#xFF0C;&#x7531;&#x4E8E;&#x53C2;&#x6570;&#x662F;&#x968F;&#x673A;&#x521D;&#x59CB;&#x5316;&#x7684;&#xFF0C;&#x68AF;&#x5EA6;&#x5F80;&#x5F80;&#x4E5F;&#x6BD4;&#x8F83;&#x5927;&#xFF0C;&#x5982;&#x679C;&#x521D;&#x59CB;&#x6B65;&#x957F;&#x4E5F;&#x5F88;&#x5927;&#xFF0C;&#x4F1A;&#x4F7F;&#x5F97;&#x8BAD;&#x7EC3;&#x4E0D;&#x7A33;&#x5B9A;</p>
<br>
<p>&#x5047;&#x8BBE;&#x9884;&#x70ED;&#x8FED;&#x4EE3;&#x6B21;&#x6570;&#x4E3A;<span class="mathjax-exps">$T&apos;$</span>&#xFF0C;&#x521D;&#x59CB;&#x6B65;&#x957F;&#x4E3A;<span class="mathjax-exps">$\eta_0$</span>&#xFF0C;&#x5219;</p>
<div>
    $$
        \begin{align*}
            \eta&apos;_t = \frac{t}{T&apos;} \eta_0, \quad t \in [T&apos;]
        \end{align*}
    $$
</div>
<p>&#x5F53;&#x9884;&#x70ED;&#x8FC7;&#x7A0B;&#x7ED3;&#x675F;&#xFF0C;&#x518D;&#x9009;&#x62E9;&#x4E00;&#x79CD;&#x6B65;&#x957F;&#x8870;&#x51CF;&#x65B9;&#x6CD5;&#x6765;&#x9010;&#x6E10;&#x964D;&#x4F4E;&#x6B65;&#x957F;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1574" class="slide " data-line="1574" data-h="9" data-v="4">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x5468;&#x671F;&#x6027;&#x6B65;&#x957F;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5728;&#x8BAD;&#x7EC3;&#x8FC7;&#x7A0B;&#x4E2D;&#x5468;&#x671F;&#x6027;&#x5730;&#x589E;&#x5927;&#x6B65;&#x957F;</p>
<br>
<p>&#x76EE;&#x7684;&#xFF1A;&#x5728;&#x8BAD;&#x7EC3;&#x8FC7;&#x7A0B;&#x4E2D;&#x6709;&#x52A9;&#x4E8E;&#x9003;&#x79BB;<span class="blue">&#x5C16;&#x9510;&#x7684;&#x5C40;&#x90E8;&#x6700;&#x5C0F;&#x503C;&#x70B9;</span>&#x548C;<span class="blue">&#x978D;&#x70B9;</span></p>
<ul>
<li>&#x5E73;&#x5766;&#x7684;&#x5C40;&#x90E8;&#x6700;&#x5C0F;&#x503C;&#x70B9;&#xFF1A;&#x9C81;&#x68D2;&#x6027;&#x597D;&#xFF0C;&#x5FAE;&#x5C0F;&#x7684;&#x53C2;&#x6570;&#x53D8;&#x52A8;&#x4E0D;&#x4F1A;&#x5BF9;&#x6A21;&#x578B;&#x6709;&#x5267;&#x70C8;&#x5F71;&#x54CD;</li>
<li>&#x5C16;&#x9510;&#x7684;&#x5C40;&#x90E8;&#x6700;&#x5C0F;&#x503C;&#x70B9;&#xFF1A;&#x9C81;&#x68D2;&#x6027;&#x5DEE;&#xFF0C;&#x5FAE;&#x5C0F;&#x7684;&#x53C2;&#x6570;&#x53D8;&#x52A8;&#x4E5F;&#x4F1A;&#x5BFC;&#x81F4;&#x6A21;&#x578B;&#x5267;&#x70C8;&#x53D8;&#x5316;</li>
</ul>
<br>
<p>&#x5468;&#x671F;&#x6027;&#x5730;&#x589E;&#x5927;&#x6B65;&#x957F;&#x867D;&#x7136;&#x4F1A;&#x77ED;&#x671F;&#x5185;&#x4F1A;&#x635F;&#x5BB3;&#x4F18;&#x5316;&#x8FC7;&#x7A0B;&#xFF0C;&#x4F46;&#x901A;&#x5E38;&#x6700;&#x7EC8;&#x4F1A;&#x6536;&#x655B;&#x5230;&#x66F4;&#x52A0;&#x7406;&#x60F3;&#x7684;&#x5C40;&#x90E8;&#x6781;&#x5C0F;&#x503C;&#x70B9;</p>
<ul>
<li><span class="blue">&#x5FAA;&#x73AF;&#x6B65;&#x957F;</span>&#xFF1A;&#x6BCF;&#x4E2A;&#x5FAA;&#x73AF;&#x5468;&#x671F;&#x7684;&#x957F;&#x5EA6;&#x90FD;&#x4E3A;<span class="mathjax-exps">$2 \Delta T$</span>&#xFF0C;&#x524D;<span class="mathjax-exps">$\Delta T$</span>&#x8F6E;&#x6B65;&#x957F;&#x7EBF;&#x6027;&#x589E;&#x5927;&#xFF0C;&#x540E;<span class="mathjax-exps">$\Delta T$</span>&#x8F6E;&#x6B65;&#x957F;&#x7EBF;&#x6027;&#x7F29;&#x5C0F;&#xFF0C;&#x7B2C;<span class="mathjax-exps">$n$</span>&#x4E2A;&#x5468;&#x671F;&#x4E2D;&#x6B65;&#x957F;&#x7684;&#x4E0A;&#x754C;&#x548C;&#x4E0B;&#x754C;&#x968F;&#x7740;<span class="mathjax-exps">$m$</span>&#x7684;&#x589E;&#x5927;&#x800C;&#x9010;&#x6E10;&#x51CF;&#x5C0F;</li>
<li><span class="blue">&#x5E26;&#x70ED;&#x91CD;&#x542F;&#x7684; SGD</span>&#xFF1A;&#x6B65;&#x957F;&#x6BCF;&#x9694;&#x4E00;&#x5B9A;&#x5468;&#x671F;&#x540E;&#x91CD;&#x65B0;&#x521D;&#x59CB;&#x5316;&#x4E3A;&#x67D0;&#x4E2A;&#x9884;&#x5148;&#x8BBE;&#x5B9A;&#x7684;&#x503C;&#xFF0C;&#x7136;&#x540E;&#x9010;&#x6E10;&#x8870;&#x51CF;&#xFF1B;&#x6BCF;&#x6B21;&#x91CD;&#x542F;&#x540E;&#x6A21;&#x578B;&#x53C2;&#x6570;&#x4E0D;&#x662F;&#x4ECE;&#x5934;&#x5F00;&#x59CB;&#x4F18;&#x5316;&#xFF0C;&#x800C;&#x662F;&#x5728;&#x91CD;&#x542F;&#x524D;&#x7684;&#x53C2;&#x6570;&#x57FA;&#x7840;&#x4E0A;&#x7EE7;&#x7EED;&#x4F18;&#x5316;</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1608" class="slide " data-line="1608" data-h="9" data-v="5">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x81EA;&#x9002;&#x5E94;&#x6B65;&#x957F;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>AdaGrad&#xFF1A;&#x7B2C;<span class="mathjax-exps">$t$</span>&#x8F6E;&#x8FED;&#x4EE3;&#x5148;&#x8BA1;&#x7B97;&#x6BCF;&#x4E2A;&#x53C2;&#x6570;<span class="blue">&#x68AF;&#x5EA6;&#x5E73;&#x65B9;&#x7684;&#x7D2F;&#x79EF;&#x503C;</span>&#xFF1A;</p>
<div>
    $$
        \begin{align*}
            \Gv_t = \sum_{\tau \in [t]} \gv_\tau \odot \gv_\tau
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\odot$</span>&#x4E3A;&#x6309;&#x5143;&#x7D20;&#x4E58;&#x79EF;&#xFF0C;<span class="mathjax-exps">$\gv_\tau \in \Rbb^{|\Wv|}$</span>&#x662F;&#x7B2C;<span class="mathjax-exps">$\tau$</span>&#x6B21;&#x8FED;&#x4EE3;&#x65F6;&#x7684;&#x68AF;&#x5EA6;</p>
<br>
<p>&#x518D;&#x5229;&#x7528;&#x7D2F;&#x79EF;&#x68AF;&#x5EA6;&#x5E73;&#x65B9;&#x505A;&#x8870;&#x51CF;&#xFF1A;</p>
<div>
    $$
        \begin{align*}
            \Wv_{t+1} \leftarrow \Wv_t - \frac{\alpha}{\sqrt{\Gv_t + \epsilon}} \odot \gv_\tau
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\alpha$</span>&#x662F;&#x521D;&#x59CB;&#x6B65;&#x957F;&#xFF0C;<span class="mathjax-exps">$\epsilon$</span>&#x662F;&#x4E3A;&#x4E86;&#x4FDD;&#x6301;&#x6570;&#x503C;&#x7A33;&#x5B9A;&#x800C;&#x8BBE;&#x7F6E;&#x7684;&#x975E;&#x5E38;&#x5C0F;&#x7684;&#x5E38;&#x6570;</p>
<br>
<p>&#x7F3A;&#x70B9;&#xFF1A;&#x7ECF;&#x8FC7;&#x4E00;&#x5B9A;&#x8F6E;&#x6570;&#x7684;&#x8FED;&#x4EE3;&#x4ECD;&#x672A;&#x627E;&#x5230;&#x6700;&#x4F18;&#x70B9;&#x65F6;&#xFF0C;&#x7531;&#x4E8E;&#x6B64;&#x65F6;&#x6B65;&#x957F;&#x5DF2;&#x7ECF;&#x975E;&#x5E38;&#x5C0F;&#xFF0C;&#x5F88;&#x96BE;&#x518D;&#x7EE7;&#x7EED;&#x627E;&#x5230;&#x6700;&#x4F18;&#x70B9;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1656" class="slide " data-line="1656" data-h="9" data-v="6">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x81EA;&#x9002;&#x5E94;&#x6B65;&#x957F;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>RMSprop&#xFF1A;&#x7B2C;<span class="mathjax-exps">$t$</span>&#x8F6E;&#x8FED;&#x4EE3;&#x5148;&#x8BA1;&#x7B97;<span class="blue">&#x68AF;&#x5EA6;&#x5E73;&#x65B9;&#x7684;&#x6307;&#x6570;&#x8870;&#x51CF;&#x79FB;&#x52A8;&#x5E73;&#x5747;</span>&#xFF1A;</p>
<div>
    $$
        \begin{align*}
            \Gv_t = \beta \Gv_{t-1} + (1 - \beta) \gv_\tau \odot \gv_\tau = (1 - \beta) \sum_{\tau \in [t]} \beta^{t - \tau} \gv_\tau \odot \gv_\tau
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\beta$</span>&#x4E3A;&#x8870;&#x51CF;&#x7387;&#xFF0C;&#x4E00;&#x822C;&#x53D6;&#x503C;<span class="mathjax-exps">$0.9$</span></p>
<br>
<p>RMSprop &#x7684;&#x66F4;&#x65B0;&#x516C;&#x5F0F;&#x4E3A;</p>
<div>
$$
    \begin{align*}
        \Wv_{t+1} \leftarrow \Wv_t - \frac{\alpha}{\sqrt{\Gv_t + \epsilon}} \odot \gv_\tau
    \end{align*}
$$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\alpha$</span>&#x901A;&#x5E38;&#x8BBE;&#x4E3A;<span class="mathjax-exps">$0.001$</span></p>
<br>
<p>&#x8F83; AdaGrad &#x7684;&#x4F18;&#x70B9;&#xFF1A;<span class="mathjax-exps">$\Gv_t$</span>&#x5E76;&#x975E;&#x5355;&#x8C03;&#x589E;&#x52A0;&#xFF0C;&#x6545;&#x6B65;&#x957F;&#x4E0D;&#x662F;&#x5355;&#x8C03;&#x8870;&#x51CF;&#xFF0C;&#x65E2;&#x53EF;&#x4EE5;&#x53D8;&#x5C0F;&#x4E5F;&#x53EF;&#x4EE5;&#x53D8;&#x5927;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1704" class="slide " data-line="1704" data-h="9" data-v="7">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x81EA;&#x9002;&#x5E94;&#x6B65;&#x957F;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>AdaDelta&#xFF1A;&#x8FDB;&#x4E00;&#x6B65;&#x5F15;&#x5165;<span class="blue">&#x53C2;&#x6570;&#x66F4;&#x65B0;&#x5DEE;&#x5E73;&#x65B9;&#x7684;&#x6307;&#x6570;&#x8870;&#x51CF;&#x79FB;&#x52A8;&#x5E73;&#x5747;</span>&#xFF1A;</p>
<div>
    $$
        \begin{align*}
            \Delta \Uv_{t-1}^2 = \beta \Delta \Uv_{t-2}^2 + (1 - \beta) \Delta \Wv_{t-1} \odot \Delta \Wv_{t-1}
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\beta$</span>&#x4E3A;&#x8870;&#x51CF;&#x7387;&#xFF0C;<span class="mathjax-exps">$\Delta \Wv_{t-1} = \Wv_t - \Wv_{t-1}$</span>&#x4E3A;&#x53C2;&#x6570;&#x66F4;&#x65B0;&#x5DEE;</p>
<br>
<p>AdaDelta &#x7684;&#x66F4;&#x65B0;&#x516C;&#x5F0F;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \Wv_{t+1} \leftarrow \Wv_t - \frac{\sqrt{\Delta \Uv_{t-1}^2 + \epsilon}}{\sqrt{\Gv_t + \epsilon}} \odot \gv_\tau
        \end{align*}
    $$
</div>
<br>
<p>&#x4F18;&#x70B9;&#xFF1A;&#x5C06; RMSprop &#x4E2D;&#x7684;&#x521D;&#x59CB;&#x6B65;&#x957F;<span class="mathjax-exps">$\alpha$</span>&#x6539;&#x4E3A;&#x52A8;&#x6001;&#x8BA1;&#x7B97;&#x7684;<span class="mathjax-exps">$\Delta \Uv_{t-1}$</span>&#xFF0C;&#x5728;&#x4E00;&#x5B9A;&#x7A0B;&#x5EA6;&#x4E0A;&#x6291;&#x5236;&#x4E86;&#x6B65;&#x957F;&#x7684;&#x6CE2;&#x52A8;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="1750" class="slide " data-line="1750" data-h="10" data-v="0">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x66F4;&#x65B0;&#x65B9;&#x5411;&#x8C03;&#x6574;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x52A8;&#x91CF;&#x6CD5;&#xFF1A;&#x8BA1;&#x7B97;&#x8D1F;&#x68AF;&#x5EA6;&#x7684;&#x201C;&#x52A0;&#x6743;&#x79FB;&#x52A8;&#x5E73;&#x5747;&#x201D;&#x4F5C;&#x4E3A;&#x53C2;&#x6570;&#x7684;&#x66F4;&#x65B0;&#x65B9;&#x5411;</p>
<div>
    $$
        \begin{align*}
            \Wv_{t+1} - \Wv_t = \Delta \Wv_t = \rho \Delta \Wv_{t-1} - \alpha \gv_t = - \alpha \sum_{\tau \in [t]} \rho^{t - \tau} \gv_\tau
        \end{align*}
    $$
</div>
<ul>
<li>&#x5F53;&#x67D0;&#x4E2A;&#x53C2;&#x6570;&#x5728;&#x6700;&#x8FD1;&#x4E00;&#x6BB5;&#x65F6;&#x95F4;&#x5185;&#x7684;&#x68AF;&#x5EA6;&#x65B9;&#x5411;&#x4E0D;&#x4E00;&#x81F4;&#x65F6;&#xFF0C;&#x5176;&#x771F;&#x5B9E;&#x7684;&#x66F4;&#x65B0;&#x5E45;&#x5EA6;&#x53D8;&#x5C0F;</li>
<li>&#x5F53;&#x67D0;&#x4E2A;&#x53C2;&#x6570;&#x5728;&#x6700;&#x8FD1;&#x4E00;&#x6BB5;&#x65F6;&#x95F4;&#x5185;&#x7684;&#x68AF;&#x5EA6;&#x65B9;&#x5411;&#x4E00;&#x81F4;&#x65F6;&#xFF0C;&#x5176;&#x771F;&#x5B9E;&#x7684;&#x66F4;&#x65B0;&#x5E45;&#x5EA6;&#x53D8;&#x5927;&#xFF0C;&#x8D77;&#x5230;&#x52A0;&#x901F;&#x4F5C;&#x7528;</li>
<li>&#x8FED;&#x4EE3;&#x521D;&#x671F;&#x68AF;&#x5EA6;&#x65B9;&#x5411;&#x6BD4;&#x8F83;&#x4E00;&#x81F4;&#xFF0C;&#x52A8;&#x91CF;&#x6CD5;&#x4F1A;&#x8D77;&#x5230;&#x52A0;&#x901F;&#x4F5C;&#x7528;&#xFF1B;&#x8FED;&#x4EE3;&#x540E;&#x671F;&#x68AF;&#x5EA6;&#x65B9;&#x5411;&#x4F1A;&#x8D8B;&#x4E8E;&#x4E0D;&#x4E00;&#x81F4;&#xFF0C;&#x5728;&#x6700;&#x4F18;&#x503C;&#x9644;&#x8FD1;&#x9707;&#x8361;&#xFF0C;&#x52A8;&#x91CF;&#x6CD5;&#x4F1A;&#x8D77;&#x5230;&#x51CF;&#x901F;&#x4F5C;&#x7528;&#xFF0C;&#x589E;&#x52A0;&#x7A33;&#x5B9A;&#x6027;</li>
</ul>
<br>
<p>Nesterov &#x52A0;&#x901F;&#x68AF;&#x5EA6;&#xFF1A;&#x6539;&#x8FDB;&#x52A8;&#x91CF;&#x6CD5;&#x7684;&#x7B2C;&#x4E8C;&#x6B65;</p>
<div>
    $$
        \begin{align*}
            \begin{cases} \widetilde{\Wv} = \Wv_t + \rho \Delta \Wv_{t-1} \\ \Wv_{t+1} = \widetilde{\Wv} - \alpha \class{yellow}{\gv_t (\Wv_t)} \end{cases} \Longrightarrow \begin{cases} \widetilde{\Wv} = \Wv_t + \rho \Delta \Wv_{t-1} \\ \Wv_{t+1} = \widetilde{\Wv} - \alpha \class{yellow}{\gv_t (\widetilde{\Wv})} \end{cases}
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1794" class="slide " data-line="1794" data-h="10" data-v="1">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x66F4;&#x65B0;&#x65B9;&#x5411;&#x8C03;&#x6574;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x52A8;&#x91CF;&#x6CD5; vs. Nesterov &#x52A0;&#x901F;&#x68AF;&#x5EA6;</p>
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" style="margin-top:5%;" height="300px">
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<p>Adam(<u>Ada</u>ptive <u>M</u>oment Estimation)&#xFF1A;&#x52A8;&#x91CF;&#x6CD5;&#x548C; RMSprop &#x7684;&#x7ED3;&#x5408;</p>
<div>
    $$
        \begin{align*}
            \Mv_t &amp; = \beta_1 \Mv_{t-1} + (1 - \beta_1) \gv_t = (1 - \beta_1) \sum_{\tau \in [t]} \beta_1^{t - \tau} \gv_\tau \\
            \Gv_t &amp; = \beta_2 \Gv_{t-1} + (1 - \beta_2) \gv_t \odot \gv_t = (1 - \beta_2) \sum_{\tau \in [t]} \beta_2^{t - \tau} \gv_\tau \odot \gv_\tau
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\beta_1$</span>&#x3001;<span class="mathjax-exps">$\beta_2$</span>&#x4E3A;&#x8870;&#x51CF;&#x7387;&#xFF0C;&#x4E00;&#x822C;&#x53D6;&#x503C;<span class="mathjax-exps">$\beta_1 = 0.9$</span>&#x3001;<span class="mathjax-exps">$\beta_2 = 0.99$</span></p>
<br>
<div>
    $$
        \begin{align*}
            \Ebb [\Mv_t] &amp; = (1 - \beta_1) \sum_{\tau \in [t]} \beta_1^{t - \tau} \Ebb [\gv_\tau] = (1 - \beta_1^t) \Ebb [\gv_\tau] \\
            \Ebb [\Gv_t] &amp; = (1 - \beta_2) \sum_{\tau \in [t]} \beta_2^{t - \tau} \Ebb [\gv_\tau \odot \gv_\tau] = (1 - \beta_2^t) \Ebb [\gv_\tau \odot \gv_\tau]
        \end{align*}
    $$
</div>
<p>&#x56E0;&#x6B64;<span class="mathjax-exps">$\widetilde{\Mv}_t = \Mv_t / (1 - \beta_1^t)$</span>&#x53EF;&#x4EE5;&#x770B;&#x4F5C;&#x4E00;&#x9636;&#x77E9;&#xFF0C;<span class="mathjax-exps">$\widetilde{\Gv}_t = \Gv_t / (1 - \beta_2^t)$</span>&#x4E3A;&#x4E8C;&#x9636;&#x77E9;</p>
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        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section><section vertical="true" data-notes lineno="1860" class="slide " data-line="1860" data-h="10" data-v="3">
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</div>
<p>Adam &#x7684;&#x66F4;&#x65B0;&#x516C;&#x5F0F;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \Wv_{t+1} \leftarrow \Wv_t - \frac{\alpha}{\sqrt{\widetilde{\Gv}_t + \epsilon}} \odot \widetilde{\Mv}_t
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\alpha$</span>&#x901A;&#x5E38;&#x8BBE;&#x4E3A;<span class="mathjax-exps">$0.001$</span>&#xFF0C;&#x4E5F;&#x53EF;&#x4EE5;&#x8FDB;&#x884C;&#x8870;&#x51CF;&#xFF0C;&#x4F8B;&#x5982;<span class="mathjax-exps">$\alpha_t = \alpha / \sqrt{t}$</span></p>
<br>
<p>&#x5982;&#x679C;&#x5C06; NAG &#x548C; RMSprop &#x7684;&#x7ED3;&#x5408;&#xFF0C;&#x5219;&#x5F97;&#x5230; Nadam</p>
<br>
<p>&#x5BF9;&#x4E8E;&#x6DF1;&#x5C42;&#x7F51;&#x7EDC;&#xFF0C;&#x5728;&#x57FA;&#x4E8E;&#x68AF;&#x5EA6;&#x4E0B;&#x964D;&#x7684;&#x8BAD;&#x7EC3;&#x8FC7;&#x7A0B;&#x4E2D;&#xFF0C;&#x9664;&#x4E86;&#x68AF;&#x5EA6;&#x6D88;&#x5931;&#xFF0C;&#x4E5F;&#x4F1A;&#x51FA;&#x73B0;&#x68AF;&#x5EA6;&#x7206;&#x70B8;&#xFF0C;&#x6B64;&#x65F6;&#x53EF;&#x8FDB;&#x884C;&#x68AF;&#x5EA6;&#x622A;&#x65AD;</p>
<ul>
<li>&#x6309;&#x503C;&#x622A;&#x65AD;&#xFF1A;<span class="mathjax-exps">$\gv_t = \max \{ \min \{ \gv_t, b \}, a \}$</span></li>
<li>&#x6309;&#x6A21;&#x622A;&#x65AD;&#xFF1A;<span class="mathjax-exps">$\gv_t = b ~ \gv_t / \| \gv_t \|$</span></li>
</ul>
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    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x8BAD;&#x7EC3;&#x6280;&#x5DE7;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x8BAD;&#x7EC3;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x6709;&#x5F88;&#x591A;&#x5947;&#x6280;&#x6DEB;&#x5DE7;</p>
<ul>
<li>&#x53C2;&#x6570;&#x521D;&#x59CB;&#x5316;</li>
<li>&#x6570;&#x636E;&#x9884;&#x5904;&#x7406;</li>
<li>&#x9010;&#x5C42;&#x5F52;&#x4E00;&#x5316;</li>
<li>&#x8D85;&#x53C2;&#x6570;&#x9009;&#x62E9;</li>
<li>&#x6743;&#x91CD;&#x8870;&#x51CF;</li>
<li>&#x63D0;&#x524D;&#x505C;&#x6B62;</li>
<li>&#x968F;&#x673A;&#x4E22;&#x5F03;</li>
<li>&#x6570;&#x636E;&#x589E;&#x5F3A;</li>
<li>Mixup</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1931" class="slide " data-line="1931" data-h="11" data-v="1">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x53C2;&#x6570;&#x521D;&#x59CB;&#x5316;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x611F;&#x77E5;&#x673A;&#x3001;&#x652F;&#x6301;&#x5411;&#x91CF;&#x673A;&#x3001;&#x5BF9;&#x6570;&#x51E0;&#x7387;&#x56DE;&#x5F52;&#x7684;<span class="mathjax-exps">$\wv$</span>&#x901A;&#x5E38;&#x521D;&#x59CB;&#x5316;&#x4E3A;&#x96F6;</p>
<br>
<p>&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7684;<span class="mathjax-exps">$\Wv$</span>&#x5982;&#x679C;&#x5168;&#x90E8;&#x521D;&#x59CB;&#x5316;&#x4E3A;&#x96F6;&#xFF0C;&#x5728;&#x7B2C;&#x4E00;&#x904D;&#x524D;&#x5411;&#x8BA1;&#x7B97;&#x65F6;&#xFF0C;&#x6240;&#x6709;&#x7684;&#x9690;&#x5C42;&#x795E;&#x7ECF;&#x5143;&#x7684;&#x6FC0;&#x6D3B;&#x503C;&#x90FD;&#x76F8;&#x540C;&#xFF0C;&#x8FD9;&#x6837;&#x4F1A;&#x5BFC;&#x81F4;&#x6DF1;&#x5C42;&#x795E;&#x7ECF;&#x5143;&#x6CA1;&#x6709;&#x533A;&#x5206;&#x6027;</p>
<br>
<p>&#x65B9;&#x6848;&#xFF1A;&#x968F;&#x673A;&#x521D;&#x59CB;&#x5316;</p>
<br>
<p>&#x7B56;&#x7565;&#xFF1A;<span class="blue">&#x4FDD;&#x6301;&#x6BCF;&#x4E2A;&#x795E;&#x7ECF;&#x5143;&#x8F93;&#x5165;&#x548C;&#x8F93;&#x51FA;&#x7684;&#x65B9;&#x5DEE;&#x4E00;&#x81F4;</span></p>
<br>
<p>&#x8003;&#x8651;&#x7B2C;<span class="mathjax-exps">$l$</span>&#x5C42;&#x7684;&#x9690;&#x85CF;&#x795E;&#x7ECF;&#x5143;<span class="mathjax-exps">$z$</span>&#xFF0C;&#x5176;&#x63A5;&#x53D7;&#x524D;&#x4E00;&#x5C42;<span class="mathjax-exps">$n_{l-1}$</span>&#x4E2A;&#x795E;&#x7ECF;&#x5143;&#x8F93;&#x51FA;<span class="mathjax-exps">$a_1, \ldots, a_{n_{l-1}}$</span>&#x4F5C;&#x4E3A;&#x8F93;&#x5165;</p>
<div>
    $$
        \begin{align*}
            z = \sum_{i \in [n_{l-1}]} w_i a_i
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="1975" class="slide " data-line="1975" data-h="11" data-v="2">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x53C2;&#x6570;&#x521D;&#x59CB;&#x5316;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5047;&#x8BBE;<span class="mathjax-exps">$w_i$</span>&#x548C;<span class="mathjax-exps">$a_i$</span>&#x7684;&#x5747;&#x503C;&#x90FD;&#x4E3A;<span class="mathjax-exps">$0$</span>&#xFF0C;&#x5E76;&#x4E14;&#x4E92;&#x76F8;&#x72EC;&#x7ACB;&#xFF0C;&#x5219;<span class="mathjax-exps">$z$</span>&#x7684;&#x65B9;&#x5DEE;&#x4E3A;</p>
<div>
$$
    \begin{align*}
        \var[z] = \sum_{i \in [n_{l-1}]} \var[w_i] \var[a_i] = n_{l-1} \var[w_i] \var[a_i]
    \end{align*}
$$
</div>
<br>
<p>&#x82E5;&#x60F3;&#x4FDD;&#x6301;&#x6BCF;&#x4E2A;&#x795E;&#x7ECF;&#x5143;&#x7684;&#x8F93;&#x5165;&#x548C;&#x8F93;&#x51FA;&#x7684;&#x65B9;&#x5DEE;&#x4E00;&#x81F4;&#xFF0C;&#x5219;&#x6709;<span class="mathjax-exps">$\var[w_i] = 1 / n_{l-1}$</span></p>
<br>
<p>&#x540C;&#x7406;&#x5728;&#x53CD;&#x5411;&#x4F20;&#x64AD;&#x4E2D;&#xFF0C;&#x82E5;&#x60F3;&#x8BEF;&#x5DEE;&#x4FE1;&#x53F7;&#x4E5F;&#x4E0D;&#x88AB;&#x653E;&#x5927;&#x6216;&#x7F29;&#x5C0F;&#xFF0C;&#x9700;&#x5C06;<span class="mathjax-exps">$w_i$</span>&#x7684;&#x65B9;&#x5DEE;&#x4FDD;&#x6301;&#x4E3A;<span class="mathjax-exps">$\var[w_i] = 1 / n_l$</span></p>
<br>
<p>&#x4E24;&#x76F8;&#x6298;&#x4E2D;&#xFF0C;&#x53EF;&#x4EE5;&#x8BBE;&#x7F6E;<span class="mathjax-exps">$\var[w_i] = 2 / (n_l + n_{l-1})$</span></p>
<ul>
<li><span class="blue">&#x6B63;&#x6001;&#x5206;&#x5E03;&#x521D;&#x59CB;&#x5316;</span>&#xFF0C;<span class="mathjax-exps">$\Ncal (0, \sqrt{2 / (n_l + n_{l-1})})$</span></li>
<li><span class="blue">&#x5747;&#x5300;&#x5206;&#x5E03;&#x521D;&#x59CB;&#x5316;</span>&#xFF0C;&#x82E5;&#x5206;&#x5E03;&#x533A;&#x95F4;&#x4E3A;<span class="mathjax-exps">$[-r,r]$</span>&#xFF0C;&#x5219;<span class="mathjax-exps">$r = \sqrt{6 / (n_l + n_{l-1})}$</span></li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="2018" class="slide " data-line="2018" data-h="11" data-v="3">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x6570;&#x636E;&#x9884;&#x5904;&#x7406;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x539F;&#x59CB;&#x6BCF;&#x4E00;&#x7EF4;&#x7279;&#x5F81;&#x7531;&#x4E8E;&#x6765;&#x6E90;&#x3001;&#x5EA6;&#x91CF;&#x4E0D;&#x540C;&#xFF0C;&#x53D6;&#x503C;&#x8303;&#x56F4;&#x53EF;&#x80FD;&#x5DEE;&#x5F02;&#x5F88;&#x5927;</p>
<br>
<p><span class="blue">&#x7F29;&#x653E;&#x5F52;&#x4E00;&#x5316;</span>&#xFF1A;&#x5C06;&#x6BCF;&#x4E00;&#x7EF4;&#x7279;&#x5F81;&#x7684;&#x53D6;&#x503C;&#x8303;&#x56F4;&#x5F52;&#x4E00;&#x5316;&#x5230;<span class="mathjax-exps">$[0, 1]$</span>&#x6216;<span class="mathjax-exps">$[&#x2212;1, 1]$</span>&#x4E4B;&#x95F4;</p>
<div>
    $$
        \begin{align*}
            \hat{x}_j = \frac{x_j - \min_i x_i}{\max_i x_i - \min_i x_i}
        \end{align*}
    $$
</div>
<br>
<p><span class="blue">&#x6807;&#x51C6;&#x5F52;&#x4E00;&#x5316;</span>&#xFF1A;&#x5C06;&#x6BCF;&#x4E00;&#x7EF4;&#x7279;&#x5F81;&#x5904;&#x7406;&#x6210;&#x7B26;&#x5408;&#x6807;&#x51C6;&#x6B63;&#x6001;&#x5206;&#x5E03;</p>
<div>
    $$
        \begin{align*}
            \hat{x}_j = \frac{x_j - \mu}{\sigma}
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\mu$</span>&#x3001;<span class="mathjax-exps">$\sigma$</span>&#x5206;&#x522B;&#x4E3A;&#x8BE5;&#x7EF4;&#x7684;&#x5747;&#x503C;&#x3001;&#x6807;&#x51C6;&#x5DEE;</p>
<br>
<p><span class="blue">&#x767D;&#x5316;</span>&#xFF1A;&#x5229;&#x7528; PCA &#x4F7F;&#x6570;&#x636E;&#x7684;&#x4E0D;&#x540C;&#x7EF4;&#x5EA6;&#x53BB;&#x76F8;&#x5173;&#xFF0C;&#x4E14;&#x6BCF;&#x4E2A;&#x7EF4;&#x5EA6;&#x7684;&#x65B9;&#x5DEE;&#x4E3A;<span class="mathjax-exps">$1$</span></p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes="&#x5176;&#x4E2D;&#x56E0;&#x4E3A;BN&#x672C;&#x8EAB;&#x5177;&#x6709;&#x5E73;&#x79FB;&#x53D8;&#x6362;&#xFF0C;&#x56E0;&#x6B64;&#x4EFF;&#x5C04;&#x53D8;&#x6362;$Wv av_l$&#x4E0D;&#x518D;&#x9700;&#x8981;&#x504F;&#x7F6E;$bv_l$" lineno="2068" class="slide " data-line="2068" data-h="11" data-v="4">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x9010;&#x5C42;&#x5F52;&#x4E00;&#x5316;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5728;&#x4F7F;&#x7528;&#x968F;&#x673A;&#x68AF;&#x5EA6;&#x4E0B;&#x964D;&#x6765;&#x8BAD;&#x7EC3;&#x7F51;&#x7EDC;&#x65F6;</p>
<ul>
<li>&#x6BCF;&#x6B21;&#x53C2;&#x6570;&#x66F4;&#x65B0;&#x90FD;&#x4F1A;&#x5BFC;&#x81F4;&#x7F51;&#x7EDC;&#x4E2D;&#x95F4;&#x6BCF;&#x4E00;&#x5C42;&#x7684;&#x8F93;&#x5165;&#x7684;&#x5206;&#x5E03;&#x53D1;&#x751F;&#x6539;&#x53D8;</li>
<li>&#x8D8A;&#x6DF1;&#x7684;&#x5C42;&#x7684;&#x8F93;&#x5165;&#x5206;&#x5E03;&#x4F1A;&#x6539;&#x53D8;&#x5F97;&#x8D8A;&#x660E;&#x663E;</li>
</ul>
<br>
<p>&#x5982;&#x679C;&#x67D0;&#x4E2A;&#x795E;&#x7ECF;&#x5C42;&#x7684;&#x8F93;&#x5165;&#x5206;&#x5E03;&#x53D1;&#x751F;&#x4E86;&#x6539;&#x53D8;&#xFF0C;&#x90A3;&#x4E48;&#x5176;&#x53C2;&#x6570;&#x9700;&#x8981;&#x91CD;&#x65B0;&#x5B66;&#x4E60;</p>
<br>
<p><span class="blue">&#x6279;&#x91CF;&#x5F52;&#x4E00;&#x5316;</span>(Batch Normalization, BN)&#xFF1A;&#x9010;&#x5C42;&#x5C06;&#x5404;&#x4E2A;&#x795E;&#x7ECF;&#x5143;<span class="mathjax-exps">$z$</span>&#x5F52;&#x4E00;&#x5316;&#x5230;&#x6807;&#x51C6;&#x6B63;&#x6001;&#x5206;&#x5E03;</p>
<div>
    $$
        \begin{align*}
            \hat{z} = \frac{z - \Ebb[z]}{\sqrt{\var [z] + \epsilon}}
        \end{align*}
    $$
</div>
<p><span class="mathjax-exps">$z$</span>&#x7684;&#x671F;&#x671B;&#x548C;&#x65B9;&#x5DEE;&#x901A;&#x5E38;&#x7528;&#x5F53;&#x524D;&#x5C0F;&#x6279;&#x91CF;&#x6837;&#x672C;&#x96C6;&#x7684;&#x5747;&#x503C;&#x548C;&#x65B9;&#x5DEE;&#x8FD1;&#x4F3C;&#x4F30;&#x8BA1;</p>
<br>
<p>&#x6279;&#x91CF;&#x5F52;&#x4E00;&#x5316;&#x64CD;&#x4F5C;&#x53EF;&#x4EE5;&#x770B;&#x4F5C;&#x4E00;&#x4E2A;&#x7279;&#x6B8A;&#x7684;&#x5C42;&#xFF0C;&#x52A0;&#x5728;&#x6BCF;&#x4E00;&#x5C42;&#x975E;&#x7EBF;&#x6027;&#x6FC0;&#x6D3B;&#x51FD;&#x6570;&#x524D;&#xFF1A;<span class="mathjax-exps">$\av_{l+1} = h(\mathrm{BN} (\Wv \av_l))$</span></p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="2113" class="slide " data-line="2113" data-h="11" data-v="5">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x9010;&#x5C42;&#x5F52;&#x4E00;&#x5316;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x6279;&#x91CF;&#x5F52;&#x4E00;&#x5316;&#xFF1A;&#x9488;&#x5BF9;&#x5355;&#x4E2A;&#x795E;&#x7ECF;&#x5143;</p>
<ul>
<li>&#x8981;&#x6C42;&#x5C0F;&#x6279;&#x91CF;&#x6837;&#x672C;&#x6570;&#x4E0D;&#x80FD;&#x592A;&#x5C0F;&#xFF0C;&#x5426;&#x5219;&#x96BE;&#x4EE5;&#x5F97;&#x5230;&#x5355;&#x4E2A;&#x795E;&#x7ECF;&#x5143;&#x8F83;&#x51C6;&#x786E;&#x7684;&#x7EDF;&#x8BA1;&#x4FE1;&#x606F;</li>
</ul>
<p>&#x5C42;&#x5F52;&#x4E00;&#x5316;&#xFF1A;&#x9488;&#x5BF9;&#x4E00;&#x5C42;&#x7684;&#x6240;&#x6709;&#x795E;&#x7ECF;&#x5143;</p>
<br>
<p>&#x8BBE;&#x5C0F;&#x6279;&#x91CF;&#x6837;&#x672C;&#x6570;&#x4E3A;<span class="mathjax-exps">$k$</span>&#xFF0C;&#x8BE5;&#x5C42;&#x795E;&#x7ECF;&#x5143;&#x6570;&#x4E3A;<span class="mathjax-exps">$n$</span></p>
<div>
    $$
        \begin{align*}
            \begin{bmatrix}
                z_{11} &amp; z_{12} &amp; \cdots &amp; z_{1n} \\
                z_{21} &amp; z_{22} &amp; \cdots &amp; z_{2n} \\
                \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\
                z_{k1} &amp; z_{k2} &amp; \cdots &amp; z_{kn} \\
            \end{bmatrix}
        \end{align*}
    $$
</div>
<br>
<ul>
<li>&#x6279;&#x91CF;&#x5F52;&#x4E00;&#x5316;&#xFF1A;&#x5BF9;&#x5217;&#x505A;&#x5F52;&#x4E00;&#x5316;</li>
<li>&#x5C42;&#x5F52;&#x4E00;&#x5316;&#xFF1A;&#x5BF9;&#x884C;&#x505A;&#x5F52;&#x4E00;&#x5316;&#xFF0C;&#x7528;&#x4E8E;&#x5C0F;&#x6279;&#x91CF;&#x6837;&#x672C;&#x6570;&#x8F83;&#x5C0F;&#x7684;&#x65F6;&#x5019;</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="2159" class="slide " data-line="2159" data-h="12" data-v="0">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x8D85;&#x53C2;&#x6570;&#x9009;&#x62E9;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x8D85;&#x53C2;&#x6570;&#x5BF9;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7684;&#x6027;&#x80FD;&#x5F71;&#x54CD;&#x5F88;&#x5927;&#xFF0C;&#x5E38;&#x89C1;&#x7684;&#x8D85;&#x53C2;&#x6570;&#x6709;</p>
<ul>
<li>&#x7F51;&#x7EDC;&#x7ED3;&#x6784;&#xFF1A;&#x795E;&#x7ECF;&#x5143;&#x4E4B;&#x95F4;&#x7684;&#x8FDE;&#x63A5;&#x5173;&#x7CFB;&#x3001;&#x5C42;&#x6570;&#x3001;&#x6BCF;&#x5C42;&#x7684;&#x795E;&#x7ECF;&#x5143;&#x6570;&#x91CF;&#x3001;&#x6FC0;&#x6D3B;&#x51FD;&#x6570;&#x7C7B;&#x578B;</li>
<li>&#x4F18;&#x5316;&#x53C2;&#x6570;&#xFF1A;&#x4F18;&#x5316;&#x65B9;&#x6CD5;&#x3001;&#x6B65;&#x957F;&#x3001;&#x5C0F;&#x6279;&#x91CF;&#x6837;&#x672C;&#x6570;</li>
<li>&#x6B63;&#x5219;&#x5316;&#x7CFB;&#x6570;</li>
</ul>
<br>
<p>&#x8D85;&#x53C2;&#x6570;&#x4F18;&#x5316;&#x5F88;&#x96BE;</p>
<ul>
<li>&#x7EC4;&#x5408;&#x4F18;&#x5316;&#x95EE;&#x9898;&#xFF0C;&#x65E0;&#x6CD5;&#x50CF;&#x4E00;&#x822C;&#x53C2;&#x6570;&#x90A3;&#x6837;&#x901A;&#x8FC7;&#x68AF;&#x5EA6;&#x4E0B;&#x964D;&#x65B9;&#x6CD5;&#x6765;&#x4F18;&#x5316;&#xFF0C;&#x4E5F;&#x6CA1;&#x6709;&#x4E00;&#x79CD;&#x901A;&#x7528;&#x6709;&#x6548;&#x7684;&#x4F18;&#x5316;&#x65B9;&#x6CD5;</li>
<li>&#x8BC4;&#x4F30;&#x4E00;&#x7EC4;&#x8D85;&#x53C2;&#x6570;&#x914D;&#x7F6E;&#x7684;&#x65F6;&#x95F4;&#x4EE3;&#x4EF7;&#x975E;&#x5E38;&#x9AD8;&#xFF0C;&#x4ECE;&#x800C;&#x5BFC;&#x81F4;&#x4E00;&#x4E9B;&#x9ED1;&#x76D2;&#x4F18;&#x5316;&#x65B9;&#x6CD5;&#xFF0C;&#x5982;&#x6F14;&#x5316;&#x7B97;&#x6CD5;&#x96BE;&#x4EE5;&#x5E94;&#x7528;</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="2190" class="slide " data-line="2190" data-h="12" data-v="1">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x7F51;&#x683C;&#x641C;&#x7D22;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5C1D;&#x8BD5;&#x6240;&#x6709;&#x7684;&#x8D85;&#x53C2;&#x6570;&#x7EC4;&#x5408;&#x6765;&#x5BFB;&#x627E;&#x4E00;&#x7EC4;&#x5408;&#x9002;&#x7684;&#x8D85;&#x53C2;&#x6570;&#x914D;&#x7F6E;</p>
<br>
<p>&#x8BBE;&#x5171;&#x6709;<span class="mathjax-exps">$K$</span>&#x4E2A;&#x8D85;&#x53C2;&#x6570;&#xFF0C;&#x7B2C;<span class="mathjax-exps">$k$</span>&#x4E2A;&#x8D85;&#x53C2;&#x6570;&#x53EF;&#x4EE5;&#x53D6;<span class="mathjax-exps">$n_k$</span>&#x4E2A;&#x503C;&#xFF0C;&#x90A3;&#x4E48;&#x7EC4;&#x5408;&#x603B;&#x6570;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            n_1 \times n_2 \times \cdots \times n_K
        \end{align*}
    $$
</div>
<br>
<p>&#x5982;&#x679C;&#x8D85;&#x53C2;&#x6570;&#x662F;&#x8FDE;&#x7EED;&#x7684;&#xFF0C;&#x53EF;&#x4EE5;&#x5C06;&#x8D85;&#x53C2;&#x6570;&#x79BB;&#x6563;&#x5316;&#xFF0C;&#x9009;&#x62E9;&#x51E0;&#x4E2A;&#x201C;&#x7ECF;&#x9A8C;&#x201D;&#x503C;&#xFF0C;&#x6BD4;&#x5982;&#x6B63;&#x5219;&#x5316;&#x7CFB;&#x6570;<span class="mathjax-exps">$\lambda$</span>&#xFF0C;&#x53EF;&#x4EE5;&#x8BBE;&#x7F6E;</p>
<div>
    $$
        \begin{align*}
            \lambda \in \{ 0.01, 0.1, 1, 10, 100 \}
        \end{align*}
    $$
</div>
<br>
<p>&#x5BF9;&#x4E8E;&#x8FDE;&#x7EED;&#x7684;&#x8D85;&#x53C2;&#x6570;&#xFF0C;&#x4E0D;&#x80FD;&#x7B80;&#x5355;&#x5730;&#x6309;&#x7B49;&#x95F4;&#x9694;&#x7684;&#x65B9;&#x5F0F;&#x79BB;&#x6563;&#x5316;&#xFF0C;&#x9700;&#x8981;&#x6839;&#x636E;&#x8D85;&#x53C2;&#x6570;&#x81EA;&#x8EAB;&#x7684;&#x7279;&#x70B9;&#x8FDB;&#x884C;&#x79BB;&#x6563;&#x5316;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="2238" class="slide " data-line="2238" data-h="12" data-v="2">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x968F;&#x673A;&#x641C;&#x7D22;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x4E0D;&#x540C;&#x8D85;&#x53C2;&#x6570;&#x5BF9;&#x6A21;&#x578B;&#x6027;&#x80FD;&#x7684;&#x5F71;&#x54CD;&#x6709;&#x5F88;&#x5927;&#x5DEE;&#x5F02;</p>
<br>
<p>&#x6709;&#x4E9B;&#x8D85;&#x53C2;&#x6570;&#x5BF9;&#x6A21;&#x578B;&#x6027;&#x80FD;&#x7684;&#x5F71;&#x54CD;&#x6709;&#x9650;&#xFF0C;&#x4F8B;&#x5982;&#x6B63;&#x5219;&#x5316;&#x7CFB;&#x6570;&#xFF1B;&#x800C;&#x6709;&#x4E9B;&#x8D85;&#x53C2;&#x6570;&#x5BF9;&#x6A21;&#x578B;&#x6027;&#x80FD;&#x5F71;&#x54CD;&#x6BD4;&#x8F83;&#x5927;&#xFF0C;&#x4F8B;&#x5982;&#x6B65;&#x957F;</p>
<br>
<p><span class="blue">&#x91C7;&#x7528;&#x7F51;&#x683C;&#x641C;&#x7D22;&#x4F1A;&#x5728;&#x4E0D;&#x91CD;&#x8981;&#x7684;&#x8D85;&#x53C2;&#x6570;&#x4E0A;&#x8FDB;&#x884C;&#x4E0D;&#x5FC5;&#x8981;&#x7684;&#x5C1D;&#x8BD5;</span></p>
<br>
<p>&#x968F;&#x673A;&#x641C;&#x7D22;&#xFF1A;&#x5BF9;&#x8D85;&#x53C2;&#x6570;&#x8FDB;&#x884C;&#x968F;&#x673A;&#x7EC4;&#x5408;&#xFF0C;&#x7136;&#x540E;&#x9009;&#x53D6;&#x4E00;&#x4E2A;&#x6027;&#x80FD;&#x6700;&#x597D;&#x7684;&#x914D;&#x7F6E;</p>
<br>
<p>&#x4F18;&#x70B9;&#xFF1A;&#x5728;&#x5B9E;&#x8DF5;&#x4E2D;&#x66F4;&#x5BB9;&#x6613;&#x5B9E;&#x73B0;&#xFF0C;&#x4E00;&#x822C;&#x4F1A;&#x6BD4;&#x7F51;&#x683C;&#x641C;&#x7D22;&#x66F4;&#x52A0;&#x6709;&#x6548;</p>
<br>
<p>&#x7F3A;&#x70B9;&#xFF1A;&#x4E0E;&#x7F51;&#x683C;&#x641C;&#x7D22;&#x4E00;&#x6837;&#xFF0C;&#x6CA1;&#x6709;&#x5229;&#x7528;&#x4E0D;&#x540C;&#x8D85;&#x53C2;&#x6570;&#x7EC4;&#x5408;&#x4E4B;&#x95F4;&#x7684;&#x76F8;&#x5173;&#x6027;&#xFF0C;&#x5373;&#x5982;&#x679C;&#x8D85;&#x53C2;&#x6570;&#x7EC4;&#x5408;&#x6BD4;&#x8F83;&#x7C7B;&#x4F3C;&#xFF0C;&#x6A21;&#x578B;&#x6027;&#x80FD;&#x4E5F;&#x4F1A;&#x6BD4;&#x8F83;&#x63A5;&#x8FD1;&#xFF0C;&#x56E0;&#x6B64;&#x8FD9;&#x4E24;&#x79CD;&#x641C;&#x7D22;&#x65B9;&#x5F0F;&#x4E00;&#x822C;&#x90FD;&#x6BD4;&#x8F83;&#x4F4E;&#x6548;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="2278" class="slide " data-line="2278" data-h="13" data-v="0">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">Bayes&#x4F18;&#x5316;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x6839;&#x636E;&#x5DF2;&#x8BD5;&#x9A8C;&#x7684;&#x8D85;&#x53C2;&#x6570;&#x7EC4;&#x5408;&#xFF0C;&#x731C;&#x6D4B;&#x53EF;&#x80FD;&#x5E26;&#x6765;&#x66F4;&#x5927;&#x6536;&#x76CA;&#x7684;&#x7EC4;&#x5408;</p>
<ul>
<li>&#x5982;&#x4F55;&#x6839;&#x636E;&#x5DF2;&#x6709;&#x8D85;&#x53C2;&#x6570;&#x7EC4;&#x5408;&#x5BF9;&#x5E94;&#x7684;&#x6A21;&#x578B;&#x6027;&#x80FD;&#xFF0C;&#x5F97;&#x5230;&#x672A;&#x77E5;&#x7EC4;&#x5408;&#x7684;&#x6A21;&#x578B;&#x6027;&#x80FD;</li>
<li>&#x5982;&#x4F55;&#x786E;&#x5B9A;&#x6536;&#x76CA;</li>
</ul>
<br>
<p>&#x7B2C;&#x4E00;&#x4E2A;&#x95EE;&#x9898;&#x901A;&#x5E38;&#x91C7;&#x7528;<span class="blue">Gauss &#x8FC7;&#x7A0B;&#x56DE;&#x5F52;</span>&#xFF0C;&#x6B64;&#x65F6;<span class="mathjax-exps">$P(y|x)$</span>&#x4E3A;&#x4E00;&#x4E2A;&#x6B63;&#x6001;&#x5206;&#x5E03;</p>
<div>
    $$
        \begin{align*}
            \Hcal = \{ \xv_i, y_i \}_{i \in [m]} \Longrightarrow P(y | \xv, \Hcal)
        \end{align*}
    $$
</div>
<br>
<p>&#x7B2C;&#x4E8C;&#x4E2A;&#x95EE;&#x9898;&#x9700;&#x5F15;&#x5165;&#x4E00;&#x4E2A;&#x6536;&#x76CA;&#x51FD;&#x6570;&#xFF0C;&#x5E38;&#x89C1;&#x7684;&#x662F;&#x671F;&#x671B;&#x6539;&#x5584;</p>
<div>
    $$
        \begin{align*}
            \int \max \{ y^\star - y, 0 \} P(y | \xv, \Hcal) \diff y
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$y^\star = \min \{ y_i, i \in [m] \}$</span>&#x662F;&#x5F53;&#x524D;&#x5DF2;&#x6709;&#x8D85;&#x53C2;&#x6570;&#x7EC4;&#x5408;&#x4E2D;&#x7684;&#x6700;&#x4F18;&#x503C;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="2327" class="slide " data-line="2327" data-h="13" data-v="1">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x52A8;&#x6001;&#x8D44;&#x6E90;&#x5206;&#x914D;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5728;&#x8D85;&#x53C2;&#x6570;&#x4F18;&#x5316;&#x4E2D;&#xFF0C;&#x5982;&#x679C;&#x53EF;&#x4EE5;&#x5728;&#x8F83;&#x65E9;&#x7684;&#x9636;&#x6BB5;&#x5C31;&#x4F30;&#x8BA1;&#x51FA;&#x4E00;&#x4E2A;&#x7EC4;&#x5408;&#x7684;&#x6548;&#x679C;&#x4F1A;&#x6BD4;&#x8F83;&#x5DEE;&#xFF0C;&#x90A3;&#x4E48;&#x53EF;&#x4EE5;&#x63D0;&#x65E9;&#x505C;&#x6B62;&#x5BF9;&#x5176;&#x7684;&#x8BC4;&#x4F30;&#xFF0C;&#x5C06;&#x66F4;&#x591A;&#x7684;&#x8BA1;&#x7B97;&#x8D44;&#x6E90;&#x7559;&#x7ED9;&#x5176;&#x5B83;&#x7EC4;&#x5408;</p>
<br>
<p>&#x9010;&#x6B21;&#x51CF;&#x534A;&#x6CD5;&#xFF1A;</p>
<ol>
<li>&#x5C06;&#x6240;&#x6709;&#x8BA1;&#x7B97;&#x8D44;&#x6E90;&#x5E73;&#x5747;&#x5206;&#x7ED9;&#x6240;&#x6709;&#x7684;&#x8D85;&#x53C2;&#x6570;&#x7EC4;&#x5408;</li>
<li>&#x540C;&#x65F6;&#x8BAD;&#x7EC3;&#x6BCF;&#x4E2A;&#x8D85;&#x53C2;&#x6570;&#x7EC4;&#x5408;&#x5BF9;&#x5E94;&#x7684;&#x7F51;&#x7EDC;&#x4E00;&#x6BB5;&#x65F6;&#x95F4;</li>
<li>&#x4FDD;&#x7559;&#x524D;&#x4E00;&#x534A;&#x597D;&#x7684;&#x7EC4;&#x5408;&#xFF0C;&#x8F6C;&#x7B2C; 1 &#x6B65;</li>
</ol>
<br>
<p>&#x5229;&#x7528;-&#x63A2;&#x7D22;&#x4E24;&#x96BE;&#x95EE;&#x9898;&#xFF1A;</p>
<ul>
<li>&#x5982;&#x679C;&#x8D85;&#x53C2;&#x6570;&#x7EC4;&#x5408;&#x6570;&#x8D8A;&#x591A;&#xFF0C;&#x5F97;&#x5230;&#x6700;&#x4F73;&#x7EC4;&#x5408;&#x7684;&#x53EF;&#x80FD;&#x6027;&#x4E5F;&#x8D8A;&#x5927;&#xFF0C;&#x4F46;&#x6BCF;&#x4E2A;&#x7EC4;&#x5408;&#x5206;&#x5230;&#x7684;&#x8BA1;&#x7B97;&#x8D44;&#x6E90;&#x5C31;&#x8D8A;&#x5C11;&#xFF0C;&#x65E9;&#x671F;&#x7684;&#x8BC4;&#x4F30;&#x7ED3;&#x679C;&#x53EF;&#x80FD;&#x4E0D;&#x51C6;</li>
<li>&#x5982;&#x679C;&#x8D85;&#x53C2;&#x6570;&#x7EC4;&#x5408;&#x6570;&#x8D8A;&#x5C11;&#xFF0C;&#x6BCF;&#x4E2A;&#x8D85;&#x53C2;&#x6570;&#x7EC4;&#x5408;&#x7684;&#x8BC4;&#x4F30;&#x4F1A;&#x8D8A;&#x51C6;&#x786E;&#xFF0C;&#x4F46;&#x6709;&#x53EF;&#x80FD;&#x65E0;&#x6CD5;&#x5F97;&#x5230;&#x6700;&#x4F18;&#x7EC4;&#x5408;</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="2362" class="slide " data-line="2362" data-h="13" data-v="2">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x795E;&#x7ECF;&#x67B6;&#x6784;&#x641C;&#x7D22;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x6DF1;&#x5EA6;&#x5B66;&#x4E60;&#xFF1A;&#x201C;&#x7279;&#x5F81;&#x5DE5;&#x7A0B;&#x201D;&#x95EE;&#x9898; <span class="mathjax-exps">$\longrightarrow$</span> &#x201C;&#x7F51;&#x7EDC;&#x67B6;&#x6784;&#x5DE5;&#x7A0B;&#x201D;&#x95EE;&#x9898;</p>
<br>
<p>&#x795E;&#x7ECF;&#x67B6;&#x6784;&#x641C;&#x7D22;(Neural Architecture Search, NAS)&#xFF1A;&#x7528;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x6765;&#x81EA;&#x52A8;&#x5B9E;&#x73B0;&#x7F51;&#x7EDC;&#x67B6;&#x6784;&#x7684;&#x8BBE;&#x8BA1;&#xFF0C;&#x76EE;&#x524D;&#x6700;&#x706B;(nei)&#x70ED;(juan)&#x7684;&#x7814;&#x7A76;&#x65B9;&#x5411;</p>
<ul>
<li>&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7684;&#x67B6;&#x6784;&#x53EF;&#x4EE5;&#x7528;&#x4E00;&#x4E2A;&#x53D8;&#x957F;&#x7684;&#x5B57;&#x7B26;&#x4E32;&#x6765;&#x63CF;&#x8FF0;</li>
<li>&#x7528;&#x53E6;&#x4E00;&#x4E2A;&#x5FAA;&#x73AF;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x6765;&#x4E0D;&#x65AD;&#x751F;&#x6210;&#x65B0;&#x7684;&#x67B6;&#x6784;&#x63CF;&#x8FF0;</li>
<li>&#x5FAA;&#x73AF;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7684;&#x8BAD;&#x7EC3;&#x91C7;&#x7528;&#x5F3A;&#x5316;&#x5B66;&#x4E60;&#x6765;&#x5B8C;&#x6210;&#xFF0C;&#x5956;&#x52B1;&#x4FE1;&#x53F7;&#x53EF;&#x4EE5;&#x4E3A;&#x751F;&#x6210;&#x7684;&#x7F51;&#x7EDC;&#x5728;&#x9A8C;&#x8BC1;&#x96C6;&#x4E0A;&#x7684;&#x6027;&#x80FD;</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="2390" class="slide " data-line="2390" data-h="14" data-v="0">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x7F51;&#x7EDC;&#x6B63;&#x5219;&#x5316;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x6743;&#x91CD;&#x8870;&#x51CF;&#xFF1A;&#x6BCF;&#x6B21;&#x66F4;&#x65B0;&#x65F6;&#x5F15;&#x5165;&#x4E00;&#x4E2A;&#x8870;&#x51CF;&#x7CFB;&#x6570;</p>
<div>
    $$
        \begin{align*}
            \Wv_{t+1} &amp; \leftarrow (1 - \beta) \Wv_t - \eta \gv_t = \Wv_t - \eta \left( \gv_t + \frac{\beta}{\eta} \Wv_t \right)
        \end{align*}
    $$
</div>
<ul>
<li>&#x5728;&#x6807;&#x51C6;&#x7684;&#x968F;&#x673A;&#x68AF;&#x5EA6;&#x4E0B;&#x964D;&#x4E2D;&#xFF0C;&#x6743;&#x91CD;&#x8870;&#x51CF;&#x7B49;&#x4EF7;&#x4E8E;<span class="mathjax-exps">$\ell_2$</span>&#x6B63;&#x5219;</li>
<li>&#x5728;&#x8F83;&#x4E3A;&#x590D;&#x6742;&#x7684;&#x4F18;&#x5316;&#x65B9;&#x6CD5;&#xFF0C;&#x4F8B;&#x5982; Adam &#x4E2D;&#xFF0C;&#x4E24;&#x8005;&#x5E76;&#x4E0D;&#x7B49;&#x4EF7;</li>
</ul>
<br>
<p>&#x63D0;&#x524D;&#x505C;&#x6B62;&#xFF1A;</p>
<ul>
<li>&#x5F15;&#x5165;&#x4E00;&#x4E2A;&#x548C;&#x8BAD;&#x7EC3;&#x96C6;&#x72EC;&#x7ACB;&#x7684;&#x6837;&#x672C;&#x96C6;&#x5408;&#xFF0C;&#x79F0;&#x4E3A;&#x9A8C;&#x8BC1;&#x96C6;(Validation Set)&#xFF0C;&#x9A8C;&#x8BC1;&#x96C6;&#x4E0A;&#x7684;&#x9519;&#x8BEF;&#x53EF;&#x89C6;&#x4E3A;&#x671F;&#x671B;&#x9519;&#x8BEF;</li>
<li>&#x5F53;&#x9A8C;&#x8BC1;&#x96C6;&#x4E0A;&#x7684;&#x9519;&#x8BEF;&#x7387;&#x4E0D;&#x518D;&#x4E0B;&#x964D;&#xFF0C;&#x5C31;&#x505C;&#x6B62;&#x8BAD;&#x7EC3;</li>
</ul>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="2428" class="slide " data-line="2428" data-h="14" data-v="1">
<div class="multi_column">
    <div class="title_hr">
        <hr class="hr_top">
        <h5 class="title">&#x968F;&#x673A;&#x4E22;&#x5F03;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5BF9;&#x6BCF;&#x4E00;&#x4E2A;&#x795E;&#x7ECF;&#x5143;&#x90FD;&#x4EE5;&#x56FA;&#x5B9A;&#x7684;&#x6982;&#x7387;<span class="mathjax-exps">$p$</span>&#x6765;&#x5224;&#x5B9A;&#x8981;&#x4E0D;&#x8981;&#x4FDD;&#x7559;</p>
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<g id="node7" class="node">
<title>22</title>
<ellipse fill="none" stroke="#586e75" cx="229" cy="-164" rx="18" ry="18"/>
</g>
<!-- 11&#45;&gt;22 -->
<g id="edge2" class="edge">
<title>11-&gt;22</title>
<path fill="none" stroke="#586e75" d="M114.9783,-219.9514C120.3404,-217.0044 126.3849,-213.7785 132,-211 160.3525,-196.9705 168.7396,-196.2141 197,-182 200.8473,-180.0649 204.8927,-177.8975 208.7637,-175.7552"/>
<polygon fill="#586e75" stroke="#586e75" points="213.2055,-173.2672 209.9428,-177.6737 211.0244,-174.489 208.8432,-175.7107 208.8432,-175.7107 208.8432,-175.7107 211.0244,-174.489 207.7436,-173.7477 213.2055,-173.2672 213.2055,-173.2672"/>
</g>
<!-- 23 -->
<g id="node8" class="node">
<title>23</title>
<ellipse fill="none" stroke="#586e75" cx="294" cy="-164" rx="18" ry="18"/>
</g>
<!-- 11&#45;&gt;23 -->
<g id="edge3" class="edge">
<title>11-&gt;23</title>
<path fill="none" stroke="#586e75" d="M114.353,-219.4645C119.774,-216.4145 126.0132,-213.2495 132,-211 187.4153,-190.1785 206.7216,-203.1824 262,-182 266.107,-180.4262 270.3247,-178.3932 274.2952,-176.2627"/>
<polygon fill="#586e75" stroke="#586e75" points="278.8211,-173.7388 275.5502,-178.1392 276.6377,-174.9564 274.4543,-176.1741 274.4543,-176.1741 274.4543,-176.1741 276.6377,-174.9564 273.3584,-174.209 278.8211,-173.7388 278.8211,-173.7388"/>
</g>
<!-- 24 -->
<g id="node9" class="node">
<title>24</title>
<ellipse fill="none" stroke="#586e75" cx="34" cy="-164" rx="18" ry="18"/>
</g>
<!-- 11&#45;&gt;24 -->
<g id="edge4" class="edge">
<title>11-&gt;24</title>
<path fill="none" stroke="#586e75" d="M86.1581,-216.1581C75.92,-205.92 61.5131,-191.5131 50.434,-180.434"/>
<polygon fill="#586e75" stroke="#586e75" points="46.8152,-176.8152 51.9418,-178.7598 48.583,-178.583 50.3508,-180.3508 50.3508,-180.3508 50.3508,-180.3508 48.583,-178.583 48.7598,-181.9418 46.8152,-176.8152 46.8152,-176.8152"/>
</g>
<!-- 25 -->
<g id="node10" class="node">
<title>25</title>
<ellipse fill="none" stroke="#586e75" cx="99" cy="-164" rx="18" ry="18"/>
</g>
<!-- 11&#45;&gt;25 -->
<g id="edge5" class="edge">
<title>11-&gt;25</title>
<path fill="none" stroke="#586e75" d="M99,-210.8939C99,-203.5688 99,-195.0213 99,-187.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="99,-182.009 101.2501,-187.009 99,-184.509 99.0001,-187.009 99.0001,-187.009 99.0001,-187.009 99,-184.509 96.7501,-187.0091 99,-182.009 99,-182.009"/>
</g>
<!-- 12 -->
<g id="node2" class="node">
<title>12</title>
<ellipse fill="none" stroke="#586e75" cx="34" cy="-229" rx="18" ry="18"/>
</g>
<!-- 12&#45;&gt;21 -->
<g id="edge6" class="edge">
<title>12-&gt;21</title>
<path fill="none" stroke="#586e75" d="M49.7854,-219.7146C54.9013,-216.8151 60.6396,-213.6771 66,-211 94.664,-196.6842 103.336,-196.3158 132,-182 135.8528,-180.0758 139.9008,-177.9135 143.7726,-175.7729"/>
<polygon fill="#586e75" stroke="#586e75" points="148.2146,-173.2854 144.9514,-177.6916 146.0333,-174.5069 143.8521,-175.7284 143.8521,-175.7284 143.8521,-175.7284 146.0333,-174.5069 142.7527,-173.7653 148.2146,-173.2854 148.2146,-173.2854"/>
</g>
<!-- 12&#45;&gt;22 -->
<g id="edge7" class="edge">
<title>12-&gt;22</title>
<path fill="none" stroke="#586e75" d="M49.1768,-219.2558C54.3552,-216.2624 60.2842,-213.185 66,-211 121.7008,-189.7075 141.2992,-203.2925 197,-182 201.1082,-180.4296 205.3266,-178.398 209.2972,-176.2681"/>
<polygon fill="#586e75" stroke="#586e75" points="213.8232,-173.7442 210.5521,-178.1445 211.6397,-174.9618 209.4563,-176.1794 209.4563,-176.1794 209.4563,-176.1794 211.6397,-174.9618 208.3604,-174.2143 213.8232,-173.7442 213.8232,-173.7442"/>
</g>
<!-- 12&#45;&gt;23 -->
<g id="edge8" class="edge">
<title>12-&gt;23</title>
<path fill="none" stroke="#586e75" d="M49.0902,-219.0166C54.2628,-216.0073 60.2103,-212.9808 66,-211 149.3182,-182.4948 178.6818,-210.5052 262,-182 266.1614,-180.5763 270.4042,-178.6124 274.3823,-176.503"/>
<polygon fill="#586e75" stroke="#586e75" points="278.9098,-173.9834 275.6349,-178.3809 276.7253,-175.1991 274.5408,-176.4148 274.5408,-176.4148 274.5408,-176.4148 276.7253,-175.1991 273.4466,-174.4487 278.9098,-173.9834 278.9098,-173.9834"/>
</g>
<!-- 12&#45;&gt;24 -->
<g id="edge9" class="edge">
<title>12-&gt;24</title>
<path fill="none" stroke="#586e75" d="M34,-210.8939C34,-203.5688 34,-195.0213 34,-187.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="34,-182.009 36.2501,-187.009 34,-184.509 34.0001,-187.009 34.0001,-187.009 34.0001,-187.009 34,-184.509 31.7501,-187.0091 34,-182.009 34,-182.009"/>
</g>
<!-- 12&#45;&gt;25 -->
<g id="edge10" class="edge">
<title>12-&gt;25</title>
<path fill="none" stroke="#586e75" d="M46.8419,-216.1581C57.08,-205.92 71.4869,-191.5131 82.566,-180.434"/>
<polygon fill="#586e75" stroke="#586e75" points="86.1848,-176.8152 84.2402,-181.9418 84.417,-178.583 82.6492,-180.3508 82.6492,-180.3508 82.6492,-180.3508 84.417,-178.583 81.0582,-178.7598 86.1848,-176.8152 86.1848,-176.8152"/>
</g>
<!-- 13 -->
<g id="node3" class="node">
<title>13</title>
<ellipse fill="none" stroke="#586e75" cx="294" cy="-229" rx="18" ry="18"/>
</g>
<!-- 13&#45;&gt;21 -->
<g id="edge11" class="edge">
<title>13-&gt;21</title>
<path fill="none" stroke="#586e75" d="M278.2055,-219.7328C273.0891,-216.8343 267.3528,-213.6923 262,-211 233.7396,-196.7859 225.3525,-196.0295 197,-182 192.8764,-179.9595 188.5212,-177.6778 184.3749,-175.4424"/>
<polygon fill="#586e75" stroke="#586e75" points="179.9783,-173.0486 185.4456,-173.4635 182.174,-174.2441 184.3696,-175.4396 184.3696,-175.4396 184.3696,-175.4396 182.174,-174.2441 183.2937,-177.4157 179.9783,-173.0486 179.9783,-173.0486"/>
</g>
<!-- 13&#45;&gt;22 -->
<g id="edge12" class="edge">
<title>13-&gt;22</title>
<path fill="none" stroke="#586e75" d="M281.1581,-216.1581C270.92,-205.92 256.5131,-191.5131 245.434,-180.434"/>
<polygon fill="#586e75" stroke="#586e75" points="241.8152,-176.8152 246.9418,-178.7598 243.583,-178.583 245.3508,-180.3508 245.3508,-180.3508 245.3508,-180.3508 243.583,-178.583 243.7598,-181.9418 241.8152,-176.8152 241.8152,-176.8152"/>
</g>
<!-- 13&#45;&gt;23 -->
<g id="edge13" class="edge">
<title>13-&gt;23</title>
<path fill="none" stroke="#586e75" d="M294,-210.8939C294,-203.5688 294,-195.0213 294,-187.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="294,-182.009 296.2501,-187.009 294,-184.509 294.0001,-187.009 294.0001,-187.009 294.0001,-187.009 294,-184.509 291.7501,-187.0091 294,-182.009 294,-182.009"/>
</g>
<!-- 13&#45;&gt;24 -->
<g id="edge14" class="edge">
<title>13-&gt;24</title>
<path fill="none" stroke="#586e75" d="M278.9098,-219.0166C273.7372,-216.0073 267.7897,-212.9808 262,-211 178.6818,-182.4948 149.3182,-210.5052 66,-182 61.8386,-180.5763 57.5958,-178.6124 53.6177,-176.503"/>
<polygon fill="#586e75" stroke="#586e75" points="49.0902,-173.9834 54.5534,-174.4487 51.2747,-175.1991 53.4592,-176.4148 53.4592,-176.4148 53.4592,-176.4148 51.2747,-175.1991 52.3651,-178.3809 49.0902,-173.9834 49.0902,-173.9834"/>
</g>
<!-- 13&#45;&gt;25 -->
<g id="edge15" class="edge">
<title>13-&gt;25</title>
<path fill="none" stroke="#586e75" d="M278.8211,-219.2612C273.6425,-216.2682 267.714,-213.1896 262,-211 206.7216,-189.8176 187.4153,-202.8215 132,-182 127.6034,-180.348 123.0707,-178.2023 118.826,-175.971"/>
<polygon fill="#586e75" stroke="#586e75" points="114.353,-173.5355 119.8203,-173.9505 116.5487,-174.731 118.7443,-175.9266 118.7443,-175.9266 118.7443,-175.9266 116.5487,-174.731 117.6683,-177.9026 114.353,-173.5355 114.353,-173.5355"/>
</g>
<!-- 14 -->
<g id="node4" class="node">
<title>14</title>
<ellipse fill="none" stroke="#586e75" cx="229" cy="-229" rx="18" ry="18"/>
</g>
<!-- 14&#45;&gt;21 -->
<g id="edge16" class="edge">
<title>14-&gt;21</title>
<path fill="none" stroke="#586e75" d="M216.1581,-216.1581C205.92,-205.92 191.5131,-191.5131 180.434,-180.434"/>
<polygon fill="#586e75" stroke="#586e75" points="176.8152,-176.8152 181.9418,-178.7598 178.583,-178.583 180.3508,-180.3508 180.3508,-180.3508 180.3508,-180.3508 178.583,-178.583 178.7598,-181.9418 176.8152,-176.8152 176.8152,-176.8152"/>
</g>
<!-- 14&#45;&gt;22 -->
<g id="edge17" class="edge">
<title>14-&gt;22</title>
<path fill="none" stroke="#586e75" d="M229,-210.8939C229,-203.5688 229,-195.0213 229,-187.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="229,-182.009 231.2501,-187.009 229,-184.509 229.0001,-187.009 229.0001,-187.009 229.0001,-187.009 229,-184.509 226.7501,-187.0091 229,-182.009 229,-182.009"/>
</g>
<!-- 14&#45;&gt;23 -->
<g id="edge18" class="edge">
<title>14-&gt;23</title>
<path fill="none" stroke="#586e75" d="M241.8419,-216.1581C252.08,-205.92 266.4869,-191.5131 277.566,-180.434"/>
<polygon fill="#586e75" stroke="#586e75" points="281.1848,-176.8152 279.2402,-181.9418 279.417,-178.583 277.6492,-180.3508 277.6492,-180.3508 277.6492,-180.3508 279.417,-178.583 276.0582,-178.7598 281.1848,-176.8152 281.1848,-176.8152"/>
</g>
<!-- 14&#45;&gt;24 -->
<g id="edge19" class="edge">
<title>14-&gt;24</title>
<path fill="none" stroke="#586e75" d="M213.8232,-219.2558C208.6448,-216.2624 202.7158,-213.185 197,-211 141.2992,-189.7075 121.7008,-203.2925 66,-182 61.8918,-180.4296 57.6734,-178.398 53.7028,-176.2681"/>
<polygon fill="#586e75" stroke="#586e75" points="49.1768,-173.7442 54.6396,-174.2143 51.3603,-174.9618 53.5437,-176.1794 53.5437,-176.1794 53.5437,-176.1794 51.3603,-174.9618 52.4479,-178.1445 49.1768,-173.7442 49.1768,-173.7442"/>
</g>
<!-- 14&#45;&gt;25 -->
<g id="edge20" class="edge">
<title>14-&gt;25</title>
<path fill="none" stroke="#586e75" d="M213.2055,-219.7328C208.0891,-216.8343 202.3528,-213.6923 197,-211 168.7396,-196.7859 160.3525,-196.0295 132,-182 127.8764,-179.9595 123.5212,-177.6778 119.3749,-175.4424"/>
<polygon fill="#586e75" stroke="#586e75" points="114.9783,-173.0486 120.4456,-173.4635 117.174,-174.2441 119.3696,-175.4396 119.3696,-175.4396 119.3696,-175.4396 117.174,-174.2441 118.2937,-177.4157 114.9783,-173.0486 114.9783,-173.0486"/>
</g>
<!-- 15 -->
<g id="node5" class="node">
<title>15</title>
<ellipse fill="none" stroke="#586e75" cx="164" cy="-229" rx="18" ry="18"/>
</g>
<!-- 15&#45;&gt;21 -->
<g id="edge21" class="edge">
<title>15-&gt;21</title>
<path fill="none" stroke="#586e75" d="M164,-210.8939C164,-203.5688 164,-195.0213 164,-187.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="164,-182.009 166.2501,-187.009 164,-184.509 164.0001,-187.009 164.0001,-187.009 164.0001,-187.009 164,-184.509 161.7501,-187.0091 164,-182.009 164,-182.009"/>
</g>
<!-- 15&#45;&gt;22 -->
<g id="edge22" class="edge">
<title>15-&gt;22</title>
<path fill="none" stroke="#586e75" d="M176.8419,-216.1581C187.08,-205.92 201.4869,-191.5131 212.566,-180.434"/>
<polygon fill="#586e75" stroke="#586e75" points="216.1848,-176.8152 214.2402,-181.9418 214.417,-178.583 212.6492,-180.3508 212.6492,-180.3508 212.6492,-180.3508 214.417,-178.583 211.0582,-178.7598 216.1848,-176.8152 216.1848,-176.8152"/>
</g>
<!-- 15&#45;&gt;23 -->
<g id="edge23" class="edge">
<title>15-&gt;23</title>
<path fill="none" stroke="#586e75" d="M179.9783,-219.9514C185.3404,-217.0044 191.3849,-213.7785 197,-211 225.3525,-196.9705 233.7396,-196.2141 262,-182 265.8473,-180.0649 269.8927,-177.8975 273.7637,-175.7552"/>
<polygon fill="#586e75" stroke="#586e75" points="278.2055,-173.2672 274.9428,-177.6737 276.0244,-174.489 273.8432,-175.7107 273.8432,-175.7107 273.8432,-175.7107 276.0244,-174.489 272.7436,-173.7477 278.2055,-173.2672 278.2055,-173.2672"/>
</g>
<!-- 15&#45;&gt;24 -->
<g id="edge24" class="edge">
<title>15-&gt;24</title>
<path fill="none" stroke="#586e75" d="M148.2146,-219.7146C143.0987,-216.8151 137.3604,-213.6771 132,-211 103.336,-196.6842 94.664,-196.3158 66,-182 62.1472,-180.0758 58.0992,-177.9135 54.2274,-175.7729"/>
<polygon fill="#586e75" stroke="#586e75" points="49.7854,-173.2854 55.2473,-173.7653 51.9667,-174.5069 54.1479,-175.7284 54.1479,-175.7284 54.1479,-175.7284 51.9667,-174.5069 53.0486,-177.6916 49.7854,-173.2854 49.7854,-173.2854"/>
</g>
<!-- 15&#45;&gt;25 -->
<g id="edge25" class="edge">
<title>15-&gt;25</title>
<path fill="none" stroke="#586e75" d="M151.1581,-216.1581C140.92,-205.92 126.5131,-191.5131 115.434,-180.434"/>
<polygon fill="#586e75" stroke="#586e75" points="111.8152,-176.8152 116.9418,-178.7598 113.583,-178.583 115.3508,-180.3508 115.3508,-180.3508 115.3508,-180.3508 113.583,-178.583 113.7598,-181.9418 111.8152,-176.8152 111.8152,-176.8152"/>
</g>
<!-- 31 -->
<g id="node11" class="node">
<title>31</title>
<ellipse fill="none" stroke="#586e75" cx="164" cy="-99" rx="18" ry="18"/>
</g>
<!-- 21&#45;&gt;31 -->
<g id="edge26" class="edge">
<title>21-&gt;31</title>
<path fill="none" stroke="#586e75" d="M164,-145.8939C164,-138.5688 164,-130.0213 164,-122.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="164,-117.009 166.2501,-122.009 164,-119.509 164.0001,-122.009 164.0001,-122.009 164.0001,-122.009 164,-119.509 161.7501,-122.0091 164,-117.009 164,-117.009"/>
</g>
<!-- 32 -->
<g id="node12" class="node">
<title>32</title>
<ellipse fill="none" stroke="#586e75" cx="229" cy="-99" rx="18" ry="18"/>
</g>
<!-- 21&#45;&gt;32 -->
<g id="edge27" class="edge">
<title>21-&gt;32</title>
<path fill="none" stroke="#586e75" d="M176.8419,-151.1581C187.08,-140.92 201.4869,-126.5131 212.566,-115.434"/>
<polygon fill="#586e75" stroke="#586e75" points="216.1848,-111.8152 214.2402,-116.9418 214.417,-113.583 212.6492,-115.3508 212.6492,-115.3508 212.6492,-115.3508 214.417,-113.583 211.0582,-113.7598 216.1848,-111.8152 216.1848,-111.8152"/>
</g>
<!-- 33 -->
<g id="node13" class="node">
<title>33</title>
<ellipse fill="none" stroke="#586e75" cx="294" cy="-99" rx="18" ry="18"/>
</g>
<!-- 21&#45;&gt;33 -->
<g id="edge28" class="edge">
<title>21-&gt;33</title>
<path fill="none" stroke="#586e75" d="M179.9783,-154.9514C185.3404,-152.0044 191.3849,-148.7785 197,-146 225.3525,-131.9705 233.7396,-131.2141 262,-117 265.8473,-115.0649 269.8927,-112.8975 273.7637,-110.7552"/>
<polygon fill="#586e75" stroke="#586e75" points="278.2055,-108.2672 274.9428,-112.6737 276.0244,-109.489 273.8432,-110.7107 273.8432,-110.7107 273.8432,-110.7107 276.0244,-109.489 272.7436,-108.7477 278.2055,-108.2672 278.2055,-108.2672"/>
</g>
<!-- 34 -->
<g id="node14" class="node">
<title>34</title>
<ellipse fill="none" stroke="#586e75" cx="34" cy="-99" rx="18" ry="18"/>
</g>
<!-- 21&#45;&gt;34 -->
<g id="edge29" class="edge">
<title>21-&gt;34</title>
<path fill="none" stroke="#586e75" d="M148.2146,-154.7146C143.0987,-151.8151 137.3604,-148.6771 132,-146 103.336,-131.6842 94.664,-131.3158 66,-117 62.1472,-115.0758 58.0992,-112.9135 54.2274,-110.7729"/>
<polygon fill="#586e75" stroke="#586e75" points="49.7854,-108.2854 55.2473,-108.7653 51.9667,-109.5069 54.1479,-110.7284 54.1479,-110.7284 54.1479,-110.7284 51.9667,-109.5069 53.0486,-112.6916 49.7854,-108.2854 49.7854,-108.2854"/>
</g>
<!-- 35 -->
<g id="node15" class="node">
<title>35</title>
<ellipse fill="none" stroke="#586e75" cx="99" cy="-99" rx="18" ry="18"/>
</g>
<!-- 21&#45;&gt;35 -->
<g id="edge30" class="edge">
<title>21-&gt;35</title>
<path fill="none" stroke="#586e75" d="M151.1581,-151.1581C140.92,-140.92 126.5131,-126.5131 115.434,-115.434"/>
<polygon fill="#586e75" stroke="#586e75" points="111.8152,-111.8152 116.9418,-113.7598 113.583,-113.583 115.3508,-115.3508 115.3508,-115.3508 115.3508,-115.3508 113.583,-113.583 113.7598,-116.9418 111.8152,-111.8152 111.8152,-111.8152"/>
</g>
<!-- 22&#45;&gt;31 -->
<g id="edge31" class="edge">
<title>22-&gt;31</title>
<path fill="none" stroke="#586e75" d="M216.1581,-151.1581C205.92,-140.92 191.5131,-126.5131 180.434,-115.434"/>
<polygon fill="#586e75" stroke="#586e75" points="176.8152,-111.8152 181.9418,-113.7598 178.583,-113.583 180.3508,-115.3508 180.3508,-115.3508 180.3508,-115.3508 178.583,-113.583 178.7598,-116.9418 176.8152,-111.8152 176.8152,-111.8152"/>
</g>
<!-- 22&#45;&gt;32 -->
<g id="edge32" class="edge">
<title>22-&gt;32</title>
<path fill="none" stroke="#586e75" d="M229,-145.8939C229,-138.5688 229,-130.0213 229,-122.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="229,-117.009 231.2501,-122.009 229,-119.509 229.0001,-122.009 229.0001,-122.009 229.0001,-122.009 229,-119.509 226.7501,-122.0091 229,-117.009 229,-117.009"/>
</g>
<!-- 22&#45;&gt;33 -->
<g id="edge33" class="edge">
<title>22-&gt;33</title>
<path fill="none" stroke="#586e75" d="M241.8419,-151.1581C252.08,-140.92 266.4869,-126.5131 277.566,-115.434"/>
<polygon fill="#586e75" stroke="#586e75" points="281.1848,-111.8152 279.2402,-116.9418 279.417,-113.583 277.6492,-115.3508 277.6492,-115.3508 277.6492,-115.3508 279.417,-113.583 276.0582,-113.7598 281.1848,-111.8152 281.1848,-111.8152"/>
</g>
<!-- 22&#45;&gt;34 -->
<g id="edge34" class="edge">
<title>22-&gt;34</title>
<path fill="none" stroke="#586e75" d="M213.8232,-154.2558C208.6448,-151.2624 202.7158,-148.185 197,-146 141.2992,-124.7075 121.7008,-138.2925 66,-117 61.8918,-115.4296 57.6734,-113.398 53.7028,-111.2681"/>
<polygon fill="#586e75" stroke="#586e75" points="49.1768,-108.7442 54.6396,-109.2143 51.3603,-109.9618 53.5437,-111.1794 53.5437,-111.1794 53.5437,-111.1794 51.3603,-109.9618 52.4479,-113.1445 49.1768,-108.7442 49.1768,-108.7442"/>
</g>
<!-- 22&#45;&gt;35 -->
<g id="edge35" class="edge">
<title>22-&gt;35</title>
<path fill="none" stroke="#586e75" d="M213.2055,-154.7328C208.0891,-151.8343 202.3528,-148.6923 197,-146 168.7396,-131.7859 160.3525,-131.0295 132,-117 127.8764,-114.9595 123.5212,-112.6778 119.3749,-110.4424"/>
<polygon fill="#586e75" stroke="#586e75" points="114.9783,-108.0486 120.4456,-108.4635 117.174,-109.2441 119.3696,-110.4396 119.3696,-110.4396 119.3696,-110.4396 117.174,-109.2441 118.2937,-112.4157 114.9783,-108.0486 114.9783,-108.0486"/>
</g>
<!-- 23&#45;&gt;31 -->
<g id="edge36" class="edge">
<title>23-&gt;31</title>
<path fill="none" stroke="#586e75" d="M278.2055,-154.7328C273.0891,-151.8343 267.3528,-148.6923 262,-146 233.7396,-131.7859 225.3525,-131.0295 197,-117 192.8764,-114.9595 188.5212,-112.6778 184.3749,-110.4424"/>
<polygon fill="#586e75" stroke="#586e75" points="179.9783,-108.0486 185.4456,-108.4635 182.174,-109.2441 184.3696,-110.4396 184.3696,-110.4396 184.3696,-110.4396 182.174,-109.2441 183.2937,-112.4157 179.9783,-108.0486 179.9783,-108.0486"/>
</g>
<!-- 23&#45;&gt;32 -->
<g id="edge37" class="edge">
<title>23-&gt;32</title>
<path fill="none" stroke="#586e75" d="M281.1581,-151.1581C270.92,-140.92 256.5131,-126.5131 245.434,-115.434"/>
<polygon fill="#586e75" stroke="#586e75" points="241.8152,-111.8152 246.9418,-113.7598 243.583,-113.583 245.3508,-115.3508 245.3508,-115.3508 245.3508,-115.3508 243.583,-113.583 243.7598,-116.9418 241.8152,-111.8152 241.8152,-111.8152"/>
</g>
<!-- 23&#45;&gt;33 -->
<g id="edge38" class="edge">
<title>23-&gt;33</title>
<path fill="none" stroke="#586e75" d="M294,-145.8939C294,-138.5688 294,-130.0213 294,-122.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="294,-117.009 296.2501,-122.009 294,-119.509 294.0001,-122.009 294.0001,-122.009 294.0001,-122.009 294,-119.509 291.7501,-122.0091 294,-117.009 294,-117.009"/>
</g>
<!-- 23&#45;&gt;34 -->
<g id="edge39" class="edge">
<title>23-&gt;34</title>
<path fill="none" stroke="#586e75" d="M278.9098,-154.0166C273.7372,-151.0073 267.7897,-147.9808 262,-146 178.6818,-117.4948 149.3182,-145.5052 66,-117 61.8386,-115.5763 57.5958,-113.6124 53.6177,-111.503"/>
<polygon fill="#586e75" stroke="#586e75" points="49.0902,-108.9834 54.5534,-109.4487 51.2747,-110.1991 53.4592,-111.4148 53.4592,-111.4148 53.4592,-111.4148 51.2747,-110.1991 52.3651,-113.3809 49.0902,-108.9834 49.0902,-108.9834"/>
</g>
<!-- 23&#45;&gt;35 -->
<g id="edge40" class="edge">
<title>23-&gt;35</title>
<path fill="none" stroke="#586e75" d="M278.8211,-154.2612C273.6425,-151.2682 267.714,-148.1896 262,-146 206.7216,-124.8176 187.4153,-137.8215 132,-117 127.6034,-115.348 123.0707,-113.2023 118.826,-110.971"/>
<polygon fill="#586e75" stroke="#586e75" points="114.353,-108.5355 119.8203,-108.9505 116.5487,-109.731 118.7443,-110.9266 118.7443,-110.9266 118.7443,-110.9266 116.5487,-109.731 117.6683,-112.9026 114.353,-108.5355 114.353,-108.5355"/>
</g>
<!-- 24&#45;&gt;31 -->
<g id="edge41" class="edge">
<title>24-&gt;31</title>
<path fill="none" stroke="#586e75" d="M49.7854,-154.7146C54.9013,-151.8151 60.6396,-148.6771 66,-146 94.664,-131.6842 103.336,-131.3158 132,-117 135.8528,-115.0758 139.9008,-112.9135 143.7726,-110.7729"/>
<polygon fill="#586e75" stroke="#586e75" points="148.2146,-108.2854 144.9514,-112.6916 146.0333,-109.5069 143.8521,-110.7284 143.8521,-110.7284 143.8521,-110.7284 146.0333,-109.5069 142.7527,-108.7653 148.2146,-108.2854 148.2146,-108.2854"/>
</g>
<!-- 24&#45;&gt;32 -->
<g id="edge42" class="edge">
<title>24-&gt;32</title>
<path fill="none" stroke="#586e75" d="M49.1768,-154.2558C54.3552,-151.2624 60.2842,-148.185 66,-146 121.7008,-124.7075 141.2992,-138.2925 197,-117 201.1082,-115.4296 205.3266,-113.398 209.2972,-111.2681"/>
<polygon fill="#586e75" stroke="#586e75" points="213.8232,-108.7442 210.5521,-113.1445 211.6397,-109.9618 209.4563,-111.1794 209.4563,-111.1794 209.4563,-111.1794 211.6397,-109.9618 208.3604,-109.2143 213.8232,-108.7442 213.8232,-108.7442"/>
</g>
<!-- 24&#45;&gt;33 -->
<g id="edge43" class="edge">
<title>24-&gt;33</title>
<path fill="none" stroke="#586e75" d="M49.0902,-154.0166C54.2628,-151.0073 60.2103,-147.9808 66,-146 149.3182,-117.4948 178.6818,-145.5052 262,-117 266.1614,-115.5763 270.4042,-113.6124 274.3823,-111.503"/>
<polygon fill="#586e75" stroke="#586e75" points="278.9098,-108.9834 275.6349,-113.3809 276.7253,-110.1991 274.5408,-111.4148 274.5408,-111.4148 274.5408,-111.4148 276.7253,-110.1991 273.4466,-109.4487 278.9098,-108.9834 278.9098,-108.9834"/>
</g>
<!-- 24&#45;&gt;34 -->
<g id="edge44" class="edge">
<title>24-&gt;34</title>
<path fill="none" stroke="#586e75" d="M34,-145.8939C34,-138.5688 34,-130.0213 34,-122.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="34,-117.009 36.2501,-122.009 34,-119.509 34.0001,-122.009 34.0001,-122.009 34.0001,-122.009 34,-119.509 31.7501,-122.0091 34,-117.009 34,-117.009"/>
</g>
<!-- 24&#45;&gt;35 -->
<g id="edge45" class="edge">
<title>24-&gt;35</title>
<path fill="none" stroke="#586e75" d="M46.8419,-151.1581C57.08,-140.92 71.4869,-126.5131 82.566,-115.434"/>
<polygon fill="#586e75" stroke="#586e75" points="86.1848,-111.8152 84.2402,-116.9418 84.417,-113.583 82.6492,-115.3508 82.6492,-115.3508 82.6492,-115.3508 84.417,-113.583 81.0582,-113.7598 86.1848,-111.8152 86.1848,-111.8152"/>
</g>
<!-- 25&#45;&gt;31 -->
<g id="edge46" class="edge">
<title>25-&gt;31</title>
<path fill="none" stroke="#586e75" d="M111.8419,-151.1581C122.08,-140.92 136.4869,-126.5131 147.566,-115.434"/>
<polygon fill="#586e75" stroke="#586e75" points="151.1848,-111.8152 149.2402,-116.9418 149.417,-113.583 147.6492,-115.3508 147.6492,-115.3508 147.6492,-115.3508 149.417,-113.583 146.0582,-113.7598 151.1848,-111.8152 151.1848,-111.8152"/>
</g>
<!-- 25&#45;&gt;32 -->
<g id="edge47" class="edge">
<title>25-&gt;32</title>
<path fill="none" stroke="#586e75" d="M114.9783,-154.9514C120.3404,-152.0044 126.3849,-148.7785 132,-146 160.3525,-131.9705 168.7396,-131.2141 197,-117 200.8473,-115.0649 204.8927,-112.8975 208.7637,-110.7552"/>
<polygon fill="#586e75" stroke="#586e75" points="213.2055,-108.2672 209.9428,-112.6737 211.0244,-109.489 208.8432,-110.7107 208.8432,-110.7107 208.8432,-110.7107 211.0244,-109.489 207.7436,-108.7477 213.2055,-108.2672 213.2055,-108.2672"/>
</g>
<!-- 25&#45;&gt;33 -->
<g id="edge48" class="edge">
<title>25-&gt;33</title>
<path fill="none" stroke="#586e75" d="M114.353,-154.4645C119.774,-151.4145 126.0132,-148.2495 132,-146 187.4153,-125.1785 206.7216,-138.1824 262,-117 266.107,-115.4262 270.3247,-113.3932 274.2952,-111.2627"/>
<polygon fill="#586e75" stroke="#586e75" points="278.8211,-108.7388 275.5502,-113.1392 276.6377,-109.9564 274.4543,-111.1741 274.4543,-111.1741 274.4543,-111.1741 276.6377,-109.9564 273.3584,-109.209 278.8211,-108.7388 278.8211,-108.7388"/>
</g>
<!-- 25&#45;&gt;34 -->
<g id="edge49" class="edge">
<title>25-&gt;34</title>
<path fill="none" stroke="#586e75" d="M86.1581,-151.1581C75.92,-140.92 61.5131,-126.5131 50.434,-115.434"/>
<polygon fill="#586e75" stroke="#586e75" points="46.8152,-111.8152 51.9418,-113.7598 48.583,-113.583 50.3508,-115.3508 50.3508,-115.3508 50.3508,-115.3508 48.583,-113.583 48.7598,-116.9418 46.8152,-111.8152 46.8152,-111.8152"/>
</g>
<!-- 25&#45;&gt;35 -->
<g id="edge50" class="edge">
<title>25-&gt;35</title>
<path fill="none" stroke="#586e75" d="M99,-145.8939C99,-138.5688 99,-130.0213 99,-122.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="99,-117.009 101.2501,-122.009 99,-119.509 99.0001,-122.009 99.0001,-122.009 99.0001,-122.009 99,-119.509 96.7501,-122.0091 99,-117.009 99,-117.009"/>
</g>
<!-- 41 -->
<g id="node16" class="node">
<title>41</title>
<ellipse fill="none" stroke="#586e75" cx="164" cy="-34" rx="18" ry="18"/>
</g>
<!-- 31&#45;&gt;41 -->
<g id="edge51" class="edge">
<title>31-&gt;41</title>
<path fill="none" stroke="#586e75" d="M164,-80.8939C164,-73.5688 164,-65.0213 164,-57.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="164,-52.009 166.2501,-57.009 164,-54.509 164.0001,-57.009 164.0001,-57.009 164.0001,-57.009 164,-54.509 161.7501,-57.0091 164,-52.009 164,-52.009"/>
</g>
<!-- 32&#45;&gt;41 -->
<g id="edge52" class="edge">
<title>32-&gt;41</title>
<path fill="none" stroke="#586e75" d="M216.1581,-86.1581C205.92,-75.92 191.5131,-61.5131 180.434,-50.434"/>
<polygon fill="#586e75" stroke="#586e75" points="176.8152,-46.8152 181.9418,-48.7598 178.583,-48.583 180.3508,-50.3508 180.3508,-50.3508 180.3508,-50.3508 178.583,-48.583 178.7598,-51.9418 176.8152,-46.8152 176.8152,-46.8152"/>
</g>
<!-- 33&#45;&gt;41 -->
<g id="edge53" class="edge">
<title>33-&gt;41</title>
<path fill="none" stroke="#586e75" d="M278.1604,-89.821C273.0415,-86.9272 267.3152,-83.7657 262,-81 235.8108,-67.3727 205.0286,-52.847 185.0495,-43.6111"/>
<polygon fill="#586e75" stroke="#586e75" points="180.4607,-41.495 185.9434,-41.5456 182.7309,-42.5419 185.0012,-43.5889 185.0012,-43.5889 185.0012,-43.5889 182.7309,-42.5419 184.0589,-45.6321 180.4607,-41.495 180.4607,-41.495"/>
</g>
<!-- 34&#45;&gt;41 -->
<g id="edge54" class="edge">
<title>34-&gt;41</title>
<path fill="none" stroke="#586e75" d="M49.8396,-89.821C54.9585,-86.9272 60.6848,-83.7657 66,-81 92.1892,-67.3727 122.9714,-52.847 142.9505,-43.6111"/>
<polygon fill="#586e75" stroke="#586e75" points="147.5393,-41.495 143.9411,-45.6321 145.2691,-42.5419 142.9988,-43.5889 142.9988,-43.5889 142.9988,-43.5889 145.2691,-42.5419 142.0566,-41.5456 147.5393,-41.495 147.5393,-41.495"/>
</g>
<!-- 35&#45;&gt;41 -->
<g id="edge55" class="edge">
<title>35-&gt;41</title>
<path fill="none" stroke="#586e75" d="M111.8419,-86.1581C122.08,-75.92 136.4869,-61.5131 147.566,-50.434"/>
<polygon fill="#586e75" stroke="#586e75" points="151.1848,-46.8152 149.2402,-51.9418 149.417,-48.583 147.6492,-50.3508 147.6492,-50.3508 147.6492,-50.3508 149.417,-48.583 146.0582,-48.7598 151.1848,-46.8152 151.1848,-46.8152"/>
</g>
<!-- 51 -->
<g id="node17" class="node">
<title>51</title>
<ellipse fill="none" stroke="#586e75" cx="424" cy="-229" rx="18" ry="18"/>
</g>
<!-- 61 -->
<g id="node22" class="node">
<title>61</title>
<ellipse fill="none" stroke="#586e75" cx="489" cy="-164" rx="18" ry="18"/>
</g>
<!-- 51&#45;&gt;61 -->
<g id="edge56" class="edge">
<title>51-&gt;61</title>
<path fill="none" stroke="#586e75" d="M436.8419,-216.1581C447.08,-205.92 461.4869,-191.5131 472.566,-180.434"/>
<polygon fill="#586e75" stroke="#586e75" points="476.1848,-176.8152 474.2402,-181.9418 474.417,-178.583 472.6492,-180.3508 472.6492,-180.3508 472.6492,-180.3508 474.417,-178.583 471.0582,-178.7598 476.1848,-176.8152 476.1848,-176.8152"/>
</g>
<!-- 62 -->
<g id="node23" class="node">
<title>62</title>
<ellipse fill="none" stroke="#586e75" stroke-dasharray="5,2" cx="554" cy="-164" rx="18" ry="18"/>
</g>
<!-- 51&#45;&gt;62 -->
<g id="edge71" class="edge">
<title>51-&gt;62</title>
</g>
<!-- 63 -->
<g id="node24" class="node">
<title>63</title>
<ellipse fill="none" stroke="#586e75" stroke-dasharray="5,2" cx="619" cy="-164" rx="18" ry="18"/>
</g>
<!-- 51&#45;&gt;63 -->
<g id="edge72" class="edge">
<title>51-&gt;63</title>
</g>
<!-- 64 -->
<g id="node25" class="node">
<title>64</title>
<ellipse fill="none" stroke="#586e75" cx="359" cy="-164" rx="18" ry="18"/>
</g>
<!-- 51&#45;&gt;64 -->
<g id="edge57" class="edge">
<title>51-&gt;64</title>
<path fill="none" stroke="#586e75" d="M411.1581,-216.1581C400.92,-205.92 386.5131,-191.5131 375.434,-180.434"/>
<polygon fill="#586e75" stroke="#586e75" points="371.8152,-176.8152 376.9418,-178.7598 373.583,-178.583 375.3508,-180.3508 375.3508,-180.3508 375.3508,-180.3508 373.583,-178.583 373.7598,-181.9418 371.8152,-176.8152 371.8152,-176.8152"/>
</g>
<!-- 65 -->
<g id="node26" class="node">
<title>65</title>
<ellipse fill="none" stroke="#586e75" stroke-dasharray="5,2" cx="424" cy="-164" rx="18" ry="18"/>
</g>
<!-- 51&#45;&gt;65 -->
<g id="edge73" class="edge">
<title>51-&gt;65</title>
</g>
<!-- 52 -->
<g id="node18" class="node">
<title>52</title>
<ellipse fill="none" stroke="#586e75" cx="359" cy="-229" rx="18" ry="18"/>
</g>
<!-- 52&#45;&gt;61 -->
<g id="edge58" class="edge">
<title>52-&gt;61</title>
<path fill="none" stroke="#586e75" d="M374.9783,-219.9514C380.3404,-217.0044 386.3849,-213.7785 392,-211 420.3525,-196.9705 428.7396,-196.2141 457,-182 460.8473,-180.0649 464.8927,-177.8975 468.7637,-175.7552"/>
<polygon fill="#586e75" stroke="#586e75" points="473.2055,-173.2672 469.9428,-177.6737 471.0244,-174.489 468.8432,-175.7107 468.8432,-175.7107 468.8432,-175.7107 471.0244,-174.489 467.7436,-173.7477 473.2055,-173.2672 473.2055,-173.2672"/>
</g>
<!-- 52&#45;&gt;62 -->
<g id="edge74" class="edge">
<title>52-&gt;62</title>
</g>
<!-- 52&#45;&gt;63 -->
<g id="edge75" class="edge">
<title>52-&gt;63</title>
</g>
<!-- 52&#45;&gt;64 -->
<g id="edge59" class="edge">
<title>52-&gt;64</title>
<path fill="none" stroke="#586e75" d="M359,-210.8939C359,-203.5688 359,-195.0213 359,-187.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="359,-182.009 361.2501,-187.009 359,-184.509 359.0001,-187.009 359.0001,-187.009 359.0001,-187.009 359,-184.509 356.7501,-187.0091 359,-182.009 359,-182.009"/>
</g>
<!-- 52&#45;&gt;65 -->
<g id="edge76" class="edge">
<title>52-&gt;65</title>
</g>
<!-- 53 -->
<g id="node19" class="node">
<title>53</title>
<ellipse fill="none" stroke="#586e75" stroke-dasharray="5,2" cx="619" cy="-229" rx="18" ry="18"/>
</g>
<!-- 53&#45;&gt;61 -->
<g id="edge80" class="edge">
<title>53-&gt;61</title>
</g>
<!-- 53&#45;&gt;62 -->
<g id="edge81" class="edge">
<title>53-&gt;62</title>
</g>
<!-- 53&#45;&gt;63 -->
<g id="edge82" class="edge">
<title>53-&gt;63</title>
</g>
<!-- 53&#45;&gt;64 -->
<g id="edge83" class="edge">
<title>53-&gt;64</title>
</g>
<!-- 53&#45;&gt;65 -->
<g id="edge84" class="edge">
<title>53-&gt;65</title>
</g>
<!-- 54 -->
<g id="node20" class="node">
<title>54</title>
<ellipse fill="none" stroke="#586e75" cx="554" cy="-229" rx="18" ry="18"/>
</g>
<!-- 54&#45;&gt;61 -->
<g id="edge60" class="edge">
<title>54-&gt;61</title>
<path fill="none" stroke="#586e75" d="M541.1581,-216.1581C530.92,-205.92 516.5131,-191.5131 505.434,-180.434"/>
<polygon fill="#586e75" stroke="#586e75" points="501.8152,-176.8152 506.9418,-178.7598 503.583,-178.583 505.3508,-180.3508 505.3508,-180.3508 505.3508,-180.3508 503.583,-178.583 503.7598,-181.9418 501.8152,-176.8152 501.8152,-176.8152"/>
</g>
<!-- 54&#45;&gt;62 -->
<g id="edge77" class="edge">
<title>54-&gt;62</title>
</g>
<!-- 54&#45;&gt;63 -->
<g id="edge78" class="edge">
<title>54-&gt;63</title>
</g>
<!-- 54&#45;&gt;64 -->
<g id="edge61" class="edge">
<title>54-&gt;64</title>
<path fill="none" stroke="#586e75" d="M538.8211,-219.2612C533.6425,-216.2682 527.714,-213.1896 522,-211 466.7216,-189.8176 447.4153,-202.8215 392,-182 387.6034,-180.348 383.0707,-178.2023 378.826,-175.971"/>
<polygon fill="#586e75" stroke="#586e75" points="374.353,-173.5355 379.8203,-173.9505 376.5487,-174.731 378.7443,-175.9266 378.7443,-175.9266 378.7443,-175.9266 376.5487,-174.731 377.6683,-177.9026 374.353,-173.5355 374.353,-173.5355"/>
</g>
<!-- 54&#45;&gt;65 -->
<g id="edge79" class="edge">
<title>54-&gt;65</title>
</g>
<!-- 55 -->
<g id="node21" class="node">
<title>55</title>
<ellipse fill="none" stroke="#586e75" stroke-dasharray="5,2" cx="489" cy="-229" rx="18" ry="18"/>
</g>
<!-- 55&#45;&gt;61 -->
<g id="edge85" class="edge">
<title>55-&gt;61</title>
</g>
<!-- 55&#45;&gt;62 -->
<g id="edge86" class="edge">
<title>55-&gt;62</title>
</g>
<!-- 55&#45;&gt;63 -->
<g id="edge87" class="edge">
<title>55-&gt;63</title>
</g>
<!-- 55&#45;&gt;64 -->
<g id="edge88" class="edge">
<title>55-&gt;64</title>
</g>
<!-- 55&#45;&gt;65 -->
<g id="edge89" class="edge">
<title>55-&gt;65</title>
</g>
<!-- 71 -->
<g id="node27" class="node">
<title>71</title>
<ellipse fill="none" stroke="#586e75" cx="489" cy="-99" rx="18" ry="18"/>
</g>
<!-- 61&#45;&gt;71 -->
<g id="edge62" class="edge">
<title>61-&gt;71</title>
<path fill="none" stroke="#586e75" d="M489,-145.8939C489,-138.5688 489,-130.0213 489,-122.2449"/>
<polygon fill="#586e75" stroke="#586e75" points="489,-117.009 491.2501,-122.009 489,-119.509 489.0001,-122.009 489.0001,-122.009 489.0001,-122.009 489,-119.509 486.7501,-122.0091 489,-117.009 489,-117.009"/>
</g>
<!-- 72 -->
<g id="node28" class="node">
<title>72</title>
<ellipse fill="none" stroke="#586e75" stroke-dasharray="5,2" cx="554" cy="-99" rx="18" ry="18"/>
</g>
<!-- 61&#45;&gt;72 -->
<g id="edge90" class="edge">
<title>61-&gt;72</title>
</g>
<!-- 73 -->
<g id="node29" class="node">
<title>73</title>
<ellipse fill="none" stroke="#586e75" cx="619" cy="-99" rx="18" ry="18"/>
</g>
<!-- 61&#45;&gt;73 -->
<g id="edge63" class="edge">
<title>61-&gt;73</title>
<path fill="none" stroke="#586e75" d="M504.9783,-154.9514C510.3404,-152.0044 516.3849,-148.7785 522,-146 550.3525,-131.9705 558.7396,-131.2141 587,-117 590.8473,-115.0649 594.8927,-112.8975 598.7637,-110.7552"/>
<polygon fill="#586e75" stroke="#586e75" points="603.2055,-108.2672 599.9428,-112.6737 601.0244,-109.489 598.8432,-110.7107 598.8432,-110.7107 598.8432,-110.7107 601.0244,-109.489 597.7436,-108.7477 603.2055,-108.2672 603.2055,-108.2672"/>
</g>
<!-- 74 -->
<g id="node30" class="node">
<title>74</title>
<ellipse fill="none" stroke="#586e75" stroke-dasharray="5,2" cx="359" cy="-99" rx="18" ry="18"/>
</g>
<!-- 61&#45;&gt;74 -->
<g id="edge91" class="edge">
<title>61-&gt;74</title>
</g>
<!-- 75 -->
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</p><div class="footer">
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x7B80;&#x4ECB;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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        <hr class="hr_top">
        <h5 class="title">&#x6570;&#x636E;&#x589E;&#x5F3A;</h5>
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<p>&#x6DF1;&#x5C42;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x9700;&#x8981;&#x5927;&#x91CF;&#x7684;&#x8BAD;&#x7EC3;&#x6570;&#x636E;&#x624D;&#x80FD;&#x83B7;&#x5F97;&#x6BD4;&#x8F83;&#x7406;&#x60F3;&#x7684;&#x6548;&#x679C;</p>
<br>
<p>&#x6570;&#x636E;&#x91CF;&#x6709;&#x9650;&#x7684;&#x60C5;&#x51B5;&#x4E0B;&#xFF0C;&#x53EF;&#x4EE5;&#x901A;&#x8FC7;&#x6570;&#x636E;&#x589E;&#x5F3A;&#x6765;&#x589E;&#x52A0;&#x6570;&#x636E;&#x91CF;&#xFF0C;&#x907F;&#x514D;&#x8FC7;&#x62DF;&#x5408;</p>
<br>
<p>&#x76EE;&#x524D;&#x6570;&#x636E;&#x589E;&#x5F3A;&#x4E3B;&#x8981;&#x7528;&#x4E8E;&#x56FE;&#x50CF;&#x6570;&#x636E;&#xFF0C;&#x6587;&#x672C;&#x7B49;&#x5176;&#x5B83;&#x7C7B;&#x578B;&#x7684;&#x6570;&#x636E;&#x8FD8;&#x6CA1;&#x6709;&#x592A;&#x597D;&#x7684;&#x65B9;&#x6CD5;</p>
<br>
<p>&#x5E38;&#x89C1;&#x7684;&#x589E;&#x5F3A;&#x65B9;&#x6CD5;&#xFF1A;</p>
<ul>
<li>&#x65CB;&#x8F6C;&#xFF1A;&#x5C06;&#x56FE;&#x50CF;&#x6309;&#x987A;&#x65F6;&#x9488;&#x6216;&#x9006;&#x65F6;&#x9488;&#x65B9;&#x5411;&#x968F;&#x673A;&#x65CB;&#x8F6C;&#x4E00;&#x5B9A;&#x89D2;&#x5EA6;</li>
<li>&#x7FFB;&#x8F6C;&#xFF1A;&#x5C06;&#x56FE;&#x50CF;&#x6CBF;&#x6C34;&#x5E73;&#x6216;&#x5782;&#x76F4;&#x65B9;&#x6CD5;&#x968F;&#x673A;&#x7FFB;&#x8F6C;&#x4E00;&#x5B9A;&#x89D2;&#x5EA6;</li>
<li>&#x7F29;&#x653E;&#xFF1A;&#x5C06;&#x56FE;&#x50CF;&#x653E;&#x5927;&#x6216;&#x7F29;&#x5C0F;&#x4E00;&#x5B9A;&#x6BD4;&#x4F8B;</li>
<li>&#x5E73;&#x79FB;&#xFF1A;&#x5C06;&#x56FE;&#x50CF;&#x6CBF;&#x6C34;&#x5E73;&#x6216;&#x5782;&#x76F4;&#x65B9;&#x6CD5;&#x5E73;&#x79FB;&#x4E00;&#x5B9A;&#x6B65;&#x957F;</li>
<li>&#x52A0;&#x566A;&#x58F0;&#xFF1A;&#x52A0;&#x5165;&#x968F;&#x673A;&#x566A;&#x58F0;</li>
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<p>&#x5229;&#x7528;&#x4EFB;&#x610F;&#x4E24;&#x4E2A;&#x6837;&#x672C;<span class="mathjax-exps">$(\xv_a, y_a)$</span>&#x3001;<span class="mathjax-exps">$(\xv_b, y_b)$</span>&#x751F;&#x6210;&#x65B0;&#x6837;&#x672C;</p>
<div>
    $$
        \begin{align*}
            (\beta \xv_a + (1 - \beta) \xv_b, \beta y_a + (1 - \beta) y_b)
        \end{align*}
    $$
</div>
<br>
<p>&#x5047;&#x8BBE;&#x6A21;&#x578B;&#x5DF2;&#x7ECF;&#x6709;&#x80FD;&#x529B;&#x9884;&#x6D4B;<span class="mathjax-exps">$y_a = f(\xv_a)$</span>&#x3001;<span class="mathjax-exps">$y_b = f(\xv_b)$</span>&#xFF0C;&#x90A3;&#x4E48;&#x6B64;&#x65F6;&#x8FD8;&#x9700;&#x6EE1;&#x8DB3;</p>
<div>
    $$
        \begin{align*}
            f(\beta \xv_a + (1 - \beta) \xv_b) &amp; = \beta y_a + (1 - \beta) y_b \\
            &amp; = \beta f(\xv_a) + (1 - \beta) f(\xv_b)
        \end{align*}
    $$
</div>
<br>
<p>&#x8FD9;&#x4E2A;&#x51FD;&#x6570;&#x65B9;&#x7A0B;&#x7684;&#x89E3;&#x662F;&#x7EBF;&#x6027;&#x51FD;&#x6570;&#xFF0C;&#x5373; mixup &#x5E0C;&#x671B;&#x5B66;&#x5230;&#x7684;<span class="mathjax-exps">$f$</span>&#x662F;&#x4E00;&#x4E2A;&#x7EBF;&#x6027;&#x51FD;&#x6570;</p>
<p style="text-align:center;margin-top:2rem">&#x62AB;&#x7740;&#x6570;&#x636E;&#x589E;&#x5F3A;&#x5916;&#x8863;&#x7684;&#x6B63;&#x5219;&#x5316;&#x65B9;&#x6CD5;</p>
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        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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